GIFT  OF 
Professor  B.L.Robertson 


Engineering  Library 


^  c*  * 


HANDBOOK 

FOR 

SURVEYORS. 


MANSFIELD   MERRIMAN, 

MEMBER  OF  AMERICAN  SOCIETY  OF  CIVIL  ENGINEERS, 


JOHN   P.  BROOKS, 

PRESIDENT  OF  THE  CLARKSON  COLLEGE  OP  TECHNOLOGY. 


FIFTH  EDITION,  REVISED. 

TOTAL  ISSUE,  TEN  THOUSAND 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 

LONDON:  CHAPMAN  &  HALL,  LIMITED 

1918 


. 


ENGINEERING  LIBRARY 

Copyright,  1895, 

3Y 

MANSFIELD  MEERIMAN 

AND 
JOHN  P.  BROOKS. 


PRESS  OF 

f  j  BRAUNWORTH  &  CO* 

G/ 2O  BOOK  MANUFACTURERS 

BROOKLYN,   N.  Y. 


PREFACE  TO  FIRST  EDITION. 


This  work  is  designed  for  the  use  of  classes  in  technical 
schools,  and  also  as  a  field  book  for  surveyors.  It  is  intended 
to  embrace  in  concise  form  the  ground  that  a  student  should 
cover  in  surveying  before  taking  up  the  subject  of  railroad 
location.  Hence  it  includes  the  fundamental  theoretical  prin- 
ciples, land  and  town  surveying,  leveling  and  simple  triangula- 
tion,  and  topography.  The  attempt  has  been  made  to  discuss 
each  of  these  topics  clearly  and  concisely,  and  in  accordance 
with  the  best  modern  methods. 

The  need  of  the  volume  arose  merely  from  the  fact  that  no 
text-book  on  elementary  surveying  in  pocket-book  form  can 
now  be  found  in  the  market.  While  in  the  field  a  student  should 
have  a  book  of  tables  ever  at  hand,  and  if  these  are  combined 
with  the  text  a  double  advantage  is  often  found,  particularly 
in  adjusting  instruments  and  in  ruling  forms  for  notes. 

In  arranging  the  order  of  presentation  the  rule  has  been  as 
far  as  possible  to  proceed  from  the  simple  to  the  complex  in  a 
iaatural  order.  For  instance,  the  most  difficult  thing  in  sur- 
veying is  the  determination  of  a  true  meridian,  and  hence  in 
this  volume  it  comes  last  of  all,  although  in  most  other  books 
it  is  presented  at  an  early  stage. 

As  all  persons  likely  to  use  the  volume  have  access  to  sur- 
veying instruments,  no  illustrations  of  these  are  given.  The 
effort  has  been  made,  however,  to  set  forth  methods  of  testing 
and  comparing  instruments  more  fully  than  is  usually  done  in 
elementary  books.  As  an  instance  of  this,  attention  is  called 
to  the  determination  of  the  eccentricity  of  the  graduated  circle 
of  a  transit  given  in  Article  27. 

The  old  terms  "  latitude"  and  "  departure,"  borrowed  from 
navigation,  are  not  here  used,  but  instead  "  latitude  difference  " 
and  "longitude  difference"  are  employed,  as  is  universally 

3 


1  PREFACE, 

done  in  geodetic  surveying  ,  the  terms  "  latitude  "  and  "  longi 
tude  "  are  moreover  used  in  the  same  sense  as  in  geodesy  and 
astronomy,     That  this  method  has  advantages  the  experience 
of  many  years  of  teaching  may  bear  witness. 

The  first  field  work  done  by  a  student  is  usually  plotted  to 
a  large  scale,  and  hence  in  Chapter  IV  the  effort  is  made  to 
clearly  distinguish  between  large-scale  and  small-scale  topog- 
raphy Both  the  transit  and  the  plane-table  method  of  stadia 
work  are  presented,  but  preference  is  given  to  the  former. 
Hydrographic  and  mine  surveying  are  briefly  outlined,  the 
latter  being  with  especial  reference  to  the  practice  in  the  an- 
thracite regions  of  Pennsylvania. 

The  tables  of  natural  functions  are  given  to  five  decimal 
places,  while  logarithms  and  logarithmic  functions  are  given 
to  six  decimals.  The  old-fashioned  traverse  table  is  omitted, 
as  it  is  of  little  value  when  sines  and  cosines  are  at  hand.  The 
tables  for  stadia  reductions  are  those  computed  by  Professor 
Arthur  Winslow  for  two  minute  intervals  of  vertical  angles. 
For  assistance  in  [compiling  Tables  III,  V,  and  VI,  acknowl- 
edgments are  due  to  the  United  States  Coast  and  Geodetic 
Survey. 


NOTE  TO  FIFTH  EDITION. 

This  edition  is  mainly^characterized  by  new  tables  of  posi- 
tions of  Polaris  and  by  a  new  chart  of  lines  of  equal  magnetic 
declination,  the  copy  for  which  has  been  kindly  furnished  by 
the  U.  S.  Coast  and  Geodetic  Survey. 

A  few  minor  revisions  have  been  made  here  and  there. 
All  known  errors  have  been  corrected. 


CONTENTS. 


CHAPTER  I. 

FUNDAMENTAL  PRINCIPLES. 

ART. 

1.  Geometry  and  Trigonometry. 7 

2.  Lines,  Angles,  and  Azimuths 10 

3.  Coordinates;  Latitudes  and  Longitudes 13 

4.  Areas  of  Triangles  and  Trapezoids 15 

5.  Areas  of  Polygons =  .. 17 

6.  Computation  of  Areas 20 

7.  Division  of  Land. , 23 

8.  Inaccessible  Distances „.  25 

9.  Elevations  and  Heights ,. 27 

10.  Errors  of  Measurements . « 29 

CHAPTER  IL 

LAND  SURVEYING. 

VI.  Chains  and  Tapes ... , 82 

12.  The  Transit ..,, 35 

13.  The  Magnetic  Needle 40 

14.  Field  Work B 44 

15.  Survey  of  a  Farm .  47 

16.  Office  Work 53 

17.  Random  Lines.... , 57 

18.  Resurveys 59 

19.  Traversing , 62 

20.  United  States  Public-Land  Surveys 64 

CHAPTER  III. 

LEVELING  AND  TRIANGULATION. 

21.  The  Level 67 

22.  Adjustments  of  the  Level 68 

23.  Comparison  of  Levels.. ...  70 

24.  Leveling 73 

25.  Contours  and  Profiles 75 

26.  Adjustments  of  the  Transit 78 

27.  Comparison  of  Transits 81 

5 


6  CONTENTS. 

AET.  PAGE 

28.  Standard  Tapes , 84 

29o  Base  Lines.   - 87 

80.  Triangulation  Work 90 

CHAPTER  IV. 

TOPOGRAPHIC  SURVEYING. 

81.  Large-Scale  Topography ,.    94 

32.  Small-Scale  Topography 98 

33.  Theory  of  the  Stadia 100 

34.  Stadia  Reductions.... 104 

35.  Field  Work  with  the  Stadia 107 

36.  Office  Work 110 

37.  The  Plane  Table 112 

38.  Hydrographic  Surveying 115 

39.  Mine  Surveying 119 

40.  The  True  Meridian 154 

41.  Isogonic  Chart  of  United  States  for  1915 123 

42.  Azimuth  by  Altitude  of  the  Sun 243 

TABLES. 

I.  Natural  Sines  and  Cosines 129 

II.  Natural  Tangents  and  Cotangents 139 

Lengths  of  Circular  Arcs 151 

III.  Daily  Variation  of  the  Magnetic  Needle 152 

IV.  Degrees  of  Longitude  and  Time 153 

V.  Elongations  and  Culminations  of  Polaris 154 

VI.  Azimuths  of  Polaris  at  Elongation 156 

VII.  Metric  and  English  Measures 158 

VIII.  Length  of  Arcs  of  Latitude  and  Longitude 159 

IX.  Reduction  of  Inclined  Distances  to  the  Horizontal 160 

X.  Stadia  Reductions  for  Reading  100 161 

XL  Logarithms  of  Numbers  169 

Constant  Numbers  and  their  Logarithms 196 

XII.  Logarithmic  Sines,  Cosines,  Tangents,  and  Cotangents 197 

XIII.  Mean  Refractions 245 


A  HANDBOOK  FOR  SURVEYORS, 


CHAPTER  I. 

FUNDAMENTAL  PRINCIPLES. 
ART.  1.    GEOMETRY  AND  TRIGONOMETRY. 

Geometry  and  Surveying  were  originally  synonymous,  as 
the  etymology  of  the  former  word  indicates.  They  originated 
in  Egypt,  where  monuments  and  boundary  lines  were  annu- 
ally obliterated  by  the  inundation  of  the  Nile.  Euclid,  pro- 
fessor of  mathematics  at  Alexandria  about  250  B.C.,  wrote  a 
treatise  on  geometry  which  has  never  been  equaled  in  logical 
methods.  Geometry  furnishes  the  principles  on  which  the 
operations  of  surveying  are  founded,  whereby  line  and  angle 
measurements,  the  computation  of  areas,  and  the  construction 
of  maps  are  effected.  Arithmetic  and  Trigonometry  are  the 
tools  by  which  the  principles  of  Geometry  are  applied. 

The  following  theorems  of  plane  geometry  are  perhaps 
those  of  greatest  importance,  but  many  others  are  constantly 
used  in  the  field  practice  of  engineers  : 

If  two  straight  lines  intersect,  the  opposite  angles  are  equal. 

Straight  lines  parallel  to  the  same  straight  line  are  parallel 
to  each  other. 

The  sum  of  the  interior  angles  of  a  polygon  is  equal  to 
twice  as  many  right  angles  as  the  polygon  has  sides  minus 
four  right  angles. 

The  sum  of  the  exterior  angles  formed  by  producing  the  sides 
of  a  polygon  is  equal  to  four  right  angles. 

The  square  upon  the  hypothenuse  of  a  right-angled  triangle 
is  equal  to  the  sum  of  the  squares  upon  the  other  two  sides. 

Angles  at  the  center  of  a  circle  are  in  the  same  ratio  as  their 
intercepted  arcs. 

An  angle  at  the  circumference  of  a  circle  is  measured  by  one 
half  the  arc  intercepted  by  its  sides. 

7 


8 


FUNDAMENTAL   PRINCIPLES. 


If  the  angles  of  two  triangles  are  equal  each  to  each,  the 
homologous  sides  are  proportional  and  the  triangles  are 
similar. 

The  areas  of  similar  polygons  are  as  the  squares  of  their  ho* 
mologous  sides. 

The  area  of  a  triable  is  measured  by  one  half  the  product 
of  its  base  and  altitude.  The  area  of  a  trapezoid  is  measured 
]«y  oiie  hal>"  tLe  prudact  of  the  sum  of  its  parallel  sides  by  its 
pit'  tudo 

The  area  of  a  sector  of  a  circle  is  measured  by  one  half  the 
product  of  its  arc  and  radius. 

The  circumference  of  a  circle  is  equal  to  its  diameter  mul- 
tiplied by  3.1415927.  The  area  of  a  circle  is  equal  to  the 
square  of  its  radius  multiplied  by  3.1415927. 

Trigonometry,  or  the  solution  of  triangles  by  means  of  sines 
and  tangents  of  the  angles,  originated  in  the  thirteenth  cen- 
tury, previous  computations  having  been  made  with  chords. 
The  following  rules  for  the  solution  of  oblique  triangles  are 
here  given  for  reference,  but  it  should  be  remembered  that  no 
surveyor  can  attain  success  unless  he  is  thoroughly  conversant 
with  all  of  them  without  the  necessity  of  referring  to  a  book. 

In  any  triangle  let  a,  6,  c,  be 
the  sides  opposite  the  angles 
A,  B,  C.  These  sides  are  pro- 
portional to  the  sines  of 
opposite  angles.  The  value  of 
each  side  may  be  expressed  in 
three  ways  in  terms  of  the  other 


sides  and  angles;  thus, 


sinB 


, 
sin  0 


sin  .4         sin  G 

sin  C      ,  sin  G 


c  =  a 


./  a   .   7g 
—  V  a  +  b  — 


f  — 7   =  U  ~. ^ 

sin  A         sin  B 
Also  each  angle  may  be  expressed  as  follows  : 

a  .             &   .                           &2  -h  c2  —  i 
sm  A  =  7  sm  J5  =  -  sm  (7,     cos  A  = — — 

0  C  60C 


GEOMETRY  AND   TRIGONOMETRY. 


b    .  b   .  a5  +  c« 

sin  B  =  -  sin  A  =  -  sin  C,    cos  5  =  —  -= 
a  c  2ac 


n      c    •  c   •     r>  „      a3  +  6*  --  c* 

sin  (7  =  -  sm  A  =  T  sin  B,     cos  (7  =  -  pr-  r  --  . 
a  b  2ab 

If  A  be  made  a  right  angle  these  reduce  to  the  formulas  for 
right  triangles,  which  are  too  well  known  to  be  repeated  here. 

When  two  sides  and  their  included  angle  are  given,  as  a,  b, 
C,  then  the  formulas 

cot  A  =  —  r-xj  -  cot  (7,        cot  B  =      .a  n  -  cot  C, 
a  sm  G  b  sm  (7 

determine  A  and  #,  while  as  a  check,  A  +  B  +  C=  180°; 
the  third  side  is  then  found  from 

c  =  a  sin  (7/sin  A 

When  the  three  sides  a,  b,  c  are  given,  the  cosines  of  the 
angles  can  be  independently  computed  from  the  formulas 
above  given.  But  some  prefer  to  divide  the  triangle  into  two 
right-angled  triangles  by  dropping  a  perpendicular  from  A 
upon  the  base  a,  thus  dividing  it  into  two  segments,  a^  and  a^. 
The  sum  of  these  segments  is  a,  their  difference  is 


Let  this  difference  be  called  d;  then 


Lastly  the  angles  are  found  by 

cos  B  =  az/c,    cos  C  =  a>i/bi,    and    sin  A  =  a  sin  B/b  ; 
as  a  check  A  +  B  +  C  =  180°. 

While  the  above  expressions  are  sufficient  for  the  solution  of 
all  plane  triangles,  there  are  other  formulas  more  convenient 
for  logarithmic  computation  for  certain  special  cases.  Tables 
of  natural  functions  are  generally  used  in  ordinary  surveying, 
particularly  in  the  field,  while  logarithmic  tables  are  perhaps 
batter  for  rapid  work  in  the  office.  The  young  surveyor 
should  be  prepared  to  solve  triangles  quickly  and  rapidly  by 
either  method. 


10  FUNDAMENTAL  PEINCIPLES. 

In  all  kinds  of  computations  a  neat  and  orderly  arrangement 
should  be  followed,  and  it  is  recommended  that  all  problems 
given  in  these  pages,  as  well  as  those  arising  in  field  practice, 
should  be  solved  in  ink  in  a  special  book  and  be  preserved  for 
reference.  Check  computations  should  in  all  cases  be  made  ; 
this  can  be  done  by  finding  the  same  quantity  in  different 
ways,  by  computing  the  three  angles  independently  and  taking 
their  sum,  or  by  using  both  natural  functions  and  logarithmic 
tables. 

Prob.  1.  Given  a  =  227.52  feet,  b  =  168. 00  feet,  (7  =  137°  25'; 
to  compute  independently  the  angles  A  and  B. 

ART.  2.    LINES,  ANGLES,  AND  AZIMUTHS. 

The  measurement  of  a  line  consists  in  finding  how  many 
times  it  contains  the  unit  of  measure.  For  several  centuries 
the  Gunter's  chain  of  66  feet  has  been  the  English  linear  unit 
for  land  measurements  ;  it  is  divided  into  100  parts,  called 
links,  and  lengths  are  expressed  in  chains  aud  links,  the  latter 
being  written  as  decimals  of  a  chain  ;  tlms  12  chains  and  72 
links  is  12.72  chains.  Although  this  chain  is  rapidly  going  out 
of  use,  the  young  surveyor  should  be  acquainted  with  it,  since 
a  large  part  of  the  land  records  in  the  United  States  is  based 
upon  it. 

In  computing  areas  the  chain  has  the  advantage  that  square 
chains  are  easily  reduced  to  acres  by  moving  the  decimal  point 
one  place  to  the  left.  This  is  because  66  feet  X  66  feet  =  4356 
square  feet,  which  is  one  tenth  of  an  acre.  For  example,  a  rect- 
angular lot  6.48  chains  long  and  2.15  chains  wide  contains 
13.932  square  chains,  or  1.3932  acres. 

The  unit  of  linear  measure  now  generally  used  in  the  United 
States  is  the  foot.  In  measuring  lines  a  chain  100  feet  long, 
divided  into  100  links,  is  used,  and  distances  are  recorded  in 
feet,  decimals  of  a  foot  being  estimated  when  possible.  Tapes 
of  various  kinds,  with  the  foot  divided  decimally,  are  also 
used,  especially  in  cities  where  precise  measurements  are 
necessary. 

Custom  and  civil  laws  have  decided  that  the  length  of  the 


LINES,   ANGLES,    AND  AZIMUTHS.  11 

boundary  line  of  a  field  is  not  the  actual  distance  on  the  sur- 
face of  the  ground,  but  that  it  is  the  projection  of  that  dis- 
tance on  a  horizontal  plane..  In  like  manner,  the  area  of  a 
field  is  not  the  exposed  superficial  surface,  but  the  projection 
of  that  surface  on  a  horizontal  plane.  In  all  land  surveying, 
therefore,  horizontal  distances  are  to  be  measured,  and  from 
these  the  areas  are  to  be  computed. 

The  angle  between  two  boundary  lines  of  a  field  is  the 
horizontal  angle  between  their  horizontal  projections.  Angles 
are  measured  by  means  of  a  graduated  plate  which  can  be 
leveled  so  as  to  be  brought-into  a  horizontal  plane.  Although 
it  is  possible  to  make  complete  surveys  by  means  of  the  chain 
alone,  it  is  much  cheaper  to  make  a  number  of  angle  measure- 
ments to  be  used  in  connection  with  a  few  measured  linear 
distances. 

The  unit  of  angular  measure  is  the  degree,  or  the  ninetieth 
part  of  a  right  angle.  The  degree  is  divided  into  sixty  minutes 
and  the  minute  into  sixty  seconds.  In  rough  land  surveying 
the  angles  are  measured  to  the  nearest  quarter  degree,  in 
ordinary  work  to  the  nearest  minute,  and  in  triangulation  they 
are  expressed  in  seconds. 

An  arc  of  a  circle  containing  57. 3  degrees,  or  more  accurately 
57.29578  degrees,  is  equal  in  length  to  the  radius.  At  a  dis- 
tance of  1000  feet  an  angle  of  one  degree  subtends  an  arc  of 
17.453  feet,  while  an  angle  of  one  minute  subtends  0.291  feet. 
The  sine  of  one  degree  is  0.017452,  and  the  sine  of  one  minute 
is  0.000291.  Thus  for  angles 
less  than  one  degree  the  sub- 
tended arcs  may  be  taken  as  ^ 
closely  proportional  to  their  sines. 

The  angle  which  a  line  makes 
with  a  standard  line  of  refer- 
ence is  called  the  azimuth  of  the 
line.  The  standard  line  is  usu- 
usually  a  north  and  south  line,  or 
meridian.  In  land  surveying 
azimuths  are  measured  from  the  north  around  through  the  east, 


12  FUNDAMENTAL   PRINCIPLES. 

south  and  west  in  the  direction  of  motion  of  the  hands  of  a 
clock.  Thus  the  azimuth  of  the  north  point  is  0°,  of  the  east 
90°,  of  the  south  180°,  and  of  the  west  270°.  In  Fig.  2  the  azi- 
muth of  the  line  AB  is  60°,  the  azimuth  of  AC  is  150°,  the  azi- 
muth of  AD  is  250°,  and  the  azimuth  of  AH  is  290°.  When 
the  azimuths  of  two  lines  are  known,  the  angle  between  them 
is  found  by  taking  the  difference  of  the  azimuths  ;  thus  DAH 
=  290° -250°  =  40°. 

The  back  azimuth  of  a  line  is  its  azimuth  measured  at  the 
other  end  with  reference  to  a  meridian  drawn  through  that 
end.  In  plane  surveying  all  the  meridians  are  parallel,  and 
hence  the  back  azimuth  of  a  line  differs  by  180°  from  the  azi- 
muth. For  instance  in  Fig.  3  let 
the  azimuth  of  AB  be  45°,  then  the 
back  azimuth  is  225°.  In  any  case 
the  back  azimuth  of  a  line  BA  is 
the  azimuth  of  AB,  the  initial  let- 
ter indicating  the  end  where  the 
azimuth  is  measured.  In  geodetic 
surveying  the  meridians  converge 
toward  the  pole,  and  hence  the 
back  azimuth  of  a  line  differs  from 
its  azimuth  by  an  amount  slightly  greater  or  less  than  180°; 
also  the  south  is  taken  as  the  initial  point,  and  the  azimuths 
are  measured  around  through  the  west,  north,  and  east. 

When  the  interior  angles  of  a  polygon  have  been  measured 
and  also  the  azimuth  of  one  of  its  sides,  the  azimuths  of  the 
other  sides  are  easily  found.  No  special  rules  need  be  given 
for  finding  these,  for  no  error  can  occur  if  a  sketch  be  drawn 
in  each  particular  case.  For  example,  in  Fig.  3,  if  the  angle 
'iB  is  75°  and  the  azimuth  of  AB  is  45°,  then  the  azimuth  of 
BC  is  150°  ;  if  further  the  angle  C  is  40°,  then  the  azimuth 
of  CD  is  290°,  and  so  on. 

Prob.  2.  A  polygon  of  six  sides  has  the  interior  angles  A 
=  58°  24',  B  =  121°  30',  G  =  123°  30',  D  =  188°  15',  E  =  95° 
14',  F  =  133°  07'.  Compute  the  azimuth  of  each  of  the  sides 
when  the  azimuth  of  AB  is  0°  00'.  Also  when  the  azimuth 
of  £(7is0000'. 


LATITUDES  AND   LONGITUDES.  13 

ART.  3.    LATITUDES  AND  LONGITUDES. 

In  geography  the  latitude  of  a  point  is  its  angular  distance 
north  or  south  from  the  equator,  and  the  longitude  of  a  point 
is  its  angular  distance  west  or  east  from  an  assumed  meridian. 
In  plane  surveying  the  meanings  of  the  words  are  analogous, 
but  the  distances  are  measured  in  feet  from  any  two  conven- 
ient lines  of  reference  which  intersect  at  right  angles ;  one  of 
these  lines  is  generally  a  north  and  south  line  or  meridian. 
Thus  in  Fig.  4  let  SN  be  a  meridian 
and  WE  be  a  line  perpendicular  to  it. 
Let  A  and  B  be  the  ends  of  the  line  / 

AB,  and  from  each  let  perpendiculars          / 

o!4 


N 

L    ' 

\ j  " 


be  drawn  to  NS  and  WE.      Then  ai  A  w 


and  biB  are  the  latitudes,  and  a  A  and        H 

bB  are  the  longitudes  of  the  points  A         \^     | 

and  B.     Latitudes  of  points  north  of  ^ 

WE  are  regarded  as  positive,  while  Fio.  4. 

those  of  points  south  of  it  are  negative.     Longitudes  east  of 

NS  are  positive,  while  those  west  of  NS  are  negative.     Thus 

the  point  C  has  a  positive  latitude  and  a  negative  longitude. 

The  difference  of  the  latitudes  of  the  ends  of  a  line  is  called 
the  latitude  difference  of  that  line;  thus  ab  is  the  latitude 
difference  of  AB.  The  difference  of  the  longitudes  of  the 
ends  of  a  line  is  called  the  longitude  difference  of  that  line  ; 
thus  atbi  is  the  longitude  difference  of  AB.  In  general  let 
Li  and  Z2  be  the  latitudes  of  two  points,  and  M*  and  Jfa  their 
longitudes;  then  L\— L*  is  the  latitude  difference  and  Mi—M* 
is  the  longitude  difference. 

When  the  length  and  azimuth  of  a  line  are  known  its  lati- 
tude and  longitude  differences  are  found  by  multiplying  the 
length  by  the  cosine  and  sine  of  the  azimuth.  Thus,  from 
Fig.  4, 

Latitude  difference  of    AB  =     ab  =  I  cos  Z. 
Longitude  difference  of  AB  =  aj)l  —I  sin  Z.     * 

For  example,  let  the  length  of  a  line  be  457.69  feet  and  its 
azimuth  be  279°  01'  44";  then  its  latitude  difference  is  +  71.83 
feet  and  its  londtudo  difference  is  — 452  02  feet. 


14  FUNDAMENTAL   PRINCIPLES. 

When  the  latitude  L\  and  longitude  M\  of  a  point  are  known, 
as  also  the  length  and  azimuth  of  a  line  joining  that  point 
with  another,  the  latitude  L*  and  the  longitude  M*  of  the 
second  point  are 

Za  =  Li  +  I  cos  Zt  MI  =  Mi  -\- 1  sin  Z. 

The  proof   of  these    equations  is  readily  seen  from  Fig.   4, 
taking  A  as  the  first  point  and  B  as  the  second. 

The  latitude  and  longitude  of  a  line  are  often  called  coor- 
dinates, while  the  two  standard  reference  lines  8N  and  WE 
are  called  the  coordinate  axes,  and  their  intersection  0  is  known 
as  the  origin  of  coordinates.  The  latitudes  and  longitudes  of 
points  in  the  four  quadrants  formed  by  these  axes  have  the 
same  signs  as  sines  and  cosines  in  trigonometry.  It  is  usual  in 
land  surveys  to  assume  the  coordinate  axes  in  such  positions 
that  all  the  points  of  the  survey  will  fall  in  the  NE  quadrant 
where  their  latitudes  and  longitudes  are  positive.  Thus  Fig. 
5  shows  a  field  ABCD  with  the  coordinates  of  each  corner 
positive  with  respect  to  the  two  axes. 

A  line  whose  azimuth  is  known  is  often  called  a  course,  the 
word  course  implying  a  definite  direction.  Lines  or  courses 
running  northward,  or  toward  the 
top  of  the  page,  are  called  north 
courses,  while  those  that  run  south- 
ward are  south  courses;  thus  in  Fig. 
5  the  lines  DA  and  AB  are  north 
courses,  while  B  C  and  CD  are  south 
courses.  Lines  running  eastward, 
or  toward  the  right  of  the  page,  are 
called  east  courses,  while  those  run- 
ning westward  are  west  courses; 
thus  AB  and  BC  are  east  courses,  while  CD  and  DA  are  west 
courses. 

The  latitude  difference  of  a  north  course  is  positive  and  is 
called  a  northing,  while  that  of  a  south  course  is  negative  and  is 
called  a  southing;  thus  ab  is  positive,  but  be  is  negative.  The 
longitude  difference  of  an  east  course  is  positive  and  is  called  an 
easting,  while  that  of  a  west  course  is  negative  and  is  called  a 


AREAS    OF   TRIANGLES  AND   TRAPEZOIDS.          15 

westing;  thus  bid  is  positive,  but  Cidi  is  negative.  If  atten- 
tion be  paid  to  the  signs  of  the  cosines  and  sines  of  the  azi- 
muth in  making  the  computations,  the  latitude  and  longitude 
differences  will  always  come  out  with  their  proper  signs.  In 
many  books  on  surveying  the  northings  and  southings  are 
called  latitudes  instead  of  latitude  differences,  while  the  east- 
ings and  westings  are  called  departures  instead  of  longitude 
differences ;  but  the  plan  here  adopted  is  more  in  accordance 
with  the  methods  of  geodesy. 

Prob.  3.  Given  the  latitude  of  one  end  of  a  line,  as  -(-  2804.4, 
its  longitude  as  -f  4661.3,  its  length  797.2  feet,  and  its  azimuth 
115°  44'  28".  Compute  the  latitude  and  longitude  of  the  other 
end.  (Draw  a  figure  before  beginning  the  solution.) 


ART.  4.    AREAS  OF  TRIANGLES  AND  TRAPEZOIDS. 

The  areas  of  fields  are  usually  expressed  in  acres,  square 
rods,  and  square  feet,  there  being  160  square  rods  in  an  acre 
and  272£  square  feet  in  a  square  rod.  In  rough  land  surveys 
the  area  is  expressed  in  acres,  roods,  and  square  rods,  a  rood 
being  one  fourth  of  an  acre.  In  speaking  of  areas  a  square 
rod  is  usually  called  simply  a  rod. 

The  area  of  any  triangle  is  equal  to  one-half  the  product  of 
the  two  sides  into  the  sine  of  their  included  angle.  Thus,  if 
a,  b,  c,  be  the  sides  opposite  the  angles  A,  B,  0,  respectively, 
the  area  can  be  expressed  in  three  ways, 

Area  =  £  db  sin  G  =  £  ac  sin  B  =  -£  be  sin  A\ 

and  if  one  of  the  angles,  as  A,  is  a  right  angle,  the  area  is 
simply  \bc.  As  an  example,  let  a  =  22.00  chains,  c  =  13.20 
chains,  and  B  =  53°  08' ;  from  Table  I  sin  B  is  found  to  be 
0.80003,  and  then  the  area  is  116.164  square  chains,  .or  11 
acres,  98  square  rods,  and  170  square  feet. 

When  the  three  sides  of  a  triangle  have  been  measured  its 
area  may  be  found  by  the  following  rule  :  Add  together  the 
three  sides  and  take  half  their  sum,  from  the  half -sum  sub- 
tract each  side  separately,  multiply  together  the  half-sum  and 
the  three  remainders,  and  take  the  square  root  of  the  product. 


16 


FUNDAMENTAL  PRINCIPLES. 


Or,  let  a,  b,  c,  be  the  three  sides,  and  *  the  half -sum  $  (a  -f  b 
-}-c);  then 

Area  =  |/s(s— a)(s— b)(s— c). 

For  example,  let  a  =  220  feet,  6  =  176  feet,  and  c  =  132  feet ; 
then  *  =  264,  s—a  =  44,  s-b  =  88,  s-c  =  132,  and  the  area  is 
11616  square  feet,  or  42f  square  rods. 

If  the  latitudes  and  longitudes  of  the  vertices  of  a  triangle 
with  respect  to  a  meridian  ON  and  a  parallel  OE  are  given, 

the  area  of  the  triangle  is  easily 
computed,  it  being  the  difference 
between  the  area  of  a  rectangle 
and  of  three  right-angled  tri- 
angles. For  example,  let  the 
latitudes  of  the  points  A,  B,  and 
C  in  Fig.  6  be  400,  250,  and  100 
feet  respectively,  and  the  corre- 
sponding longitudes  be  500,  700,  and  80  feet.  Then  the  height 
of  the  rectangle  is  300  feet  and  its  width  is  620  feet,  which 
^ive  186,000  square  feet  for  its  area.  The  sum  of  the  areas 
of  the  three  right-angled  triangles  is  124,500  square  feet. 
Hence  the  area  of  A  B  G  is  1  acre  and  17,940  square  feet. 

The  area  of  a  trapezoid  is  equal  to  half  the  sum  of  the  par- 
allel sides  multiplied  by  its  altitude.  The  trapezoids  of  most 
common  occurrence  in  surveying  have  two  right  angles,  as  for 
instance  aABb  in  Fig.  5,  whose  area  is  \(aA  -f-  bB}ab.  In 
order  to  determine  the  area  of  an  irregular  figure  like  that  of 
ABCD  in  Fig.  7,  perpendiculars,  or  offsets,  are  sometimes 
erected  upon  the  straight  line  AD  and  their  lengths  measured 
as  well  as  their  distances  apart,  the  distances  be,  cd,  etc.,  being 


Fio.  7. 

such  that  Bci,  c^di,  etc.,  may  be    regarded    as    practically 
straight.     Then  the  total  area  is  the  sum  of  the  areas  of  the 


AREAS   OF   POLYGONS. 


17 


triangle  ABb,  and  of  the  trapezoids  bBciC,  ccidid,  etc.  This 
method  is  particularly  applicable  to  cases  where  the  lengths  of 
the  offsets  are  less  than  one  or  two  chains  and  where  great 
precision  is  not  required. 

The  area  of  any  polygon  may  be  determined  by  dividing  it 
into  triangles.  Fig.  8  shows  two 
ways  of  thus  dividing  a  six-sided 
field,  and  many  others  are  pos- 
sible. In  practice  it  is  more  ad- 
vantageous  to  measure  a  number 
of  angles  and  a  few  sides,  rather 
than  all  the  sides  of  all  the  tri- 
angles. But  a  better  method  for  computing  the  area  of  a 
polygon  is  by  means  of  trapezoids,  as  explained  in  the  next 
article. 

Prob.  4.  Compute  the  area  of  the  first  diagram  in  Fig.  8 
from  the  following  data  :  AB  =  317.8  feet,  BF  —  284.3  feet, 
FA  =  250.5  feet,  F0  =  512.7  feet,  FD  -  510.0  feet,  DEF- 
90°  00',  EFD  =  69°  45',  DFO=  61°  12',  CFB  =  49°  30'. 


B 


B 


ART.  5.    AREAS  OF  POLYGONS. 

To  determine  the  area  of  a  polygonal  field  it  is  customary  to 
measure  the  length  of  each  side  and  each  of  the  interior  angles. 
The  azimuth  of  one  side  is  also  either  determined  or  assumed  ; 
then  by  Art.  2  the  azimuth  of  each  of  the  other  sides  is  readily 
found.  Let  ABGDEA  in  Fig.  9  be  a  field  in  which  the  length 
and  azimuth  of  each  side  is  known. 
It  is  required  to  deduce  a  method 
for  computing  the  area. 

Let  a  meridian  be  drawn  through 
the  most  westerly  corner  of  the 
field,  and  from  each  of  the  other 
corners  let  perpendiculars  Bbt  Cc, 
Dd,  and  J£e"be  drawn  to  it;  these 
are  the  longitudes  of  the  corners 
(Art,  3).  Then  the  area  of  the 
field  is  equal  to  the  area  bBCDd  minus  the  areas  AbB  and 


L8  FUNDAMENTAL   PRINCIPLES. 

AEDd.     The  first  area  is  formed  by  the  two  trapezoids  bBCc 
and  cCDd,  the  second  is  the  triangle  AbB,  while  the  third  is 
formed  by  the  triangle  AEe  and  the  trapezoid  eEDd.     Hence 
Area  =  $(bB  +  cC)bc  +  i(cC+dD)cd 

-  IbB .  Ab  -  \eE.  eA  -  \(dD  +  eE)de, 
and  the  double  area  of  the  field  is 

2  Area  =  (bB  +  cC)bc  +  (cO+  dD)cd  -  bB .  Ab 

-eE.eA-  (dD  +  eE)det 

and  it  has  been  shown  in  Art.  3  how  all  the  quantities  in  this 
expression  can  be  computed. 

The  longitude  of  a  point  is  its  distance  from  the  meridian 
(Art.  3);  thus  bB  and  cC  are  the  longitudes  of  the  points  B  and 
C.  The  longitude  of  a  line  or  course  may  now  be  defined  to 
be  the  longitude  of  its  middle  point,  thus  \(bB-\-cG)  is  the 
longitude  of  the  course  BG.  Hence  bB-\-cC  is  the  double 
longitude  of  BC,  or  the  double  longitude  of  any  course  is  the 
sum  of  the  longitudes  of  its  ends. 

Inspection  of  the  above  expression  for  the  double  area  of  * 
field  shows  two  facts  :  First,  that  the  double  area  is  the  differ- 
ence of  two  quantities,  one  being  the  sum  of  the  areas  of  the 
trapezoids  included  between  the  south  courses  and  the  meridian, 
while  the  other  is  the  sum  of  the  areas  of  the  trapezoids  and 
triangles  included  between  the  north  courses  and  the  meridian. 
Second,  that  each  of  these  areas  is  the  product  of  the  double 
longitude  of  a  course  by  its  latitude  difference.  Hence  let 
Si ,  $2 ,  etc. ,  be  the  double  longitudes  of  the  south  courses  and 
Si  ,  Sz  ,  etc.,  their  southings,  and  let  ffii  ,  -ZVa ,  etc.,  be  the 
double  longitudes  of  the  north  courses,  and  n\ ,  n^ ,  etc.,  their 
northings  ;  then 

2  Area  =  SiSi  -\-  JS^  -J-  etc.  —  N\UI  —  Ntfi*  —  etc. 
gives  a  general  rule  for  computing  the  area  of  any  polygonal 
field.     The  areas  SiSi ,  /S^s ,  etc.,  are  often  called  south  areas, 
while  the  others  are  called  north  areas. 

The  northings  and  southings  of  each  course  having  bee  a 
computed  by  Art.  3,  as  also  the  eastings  and  westings,  it  only 
remains  to  find  the  double  longitudes.  For  the  first  course 


AREAS  OF  POLYGON'S.  19 

AB  the  double  longitude  is  its  easting  W.  For  the  second 
course  BC  the  double  longitude  is  bB-\-cC,  that  is,  bB  -f  bB  -f- 
bi  C.  For  the  third  course  CD  the  double  longitude  is  c  C  +  dD, 
that  is,  bB-\- cC+biC  —  CdL.  In  general  the  following  rule 
will  be  useful : 

The  double  longitude  of  any  course  is  equal  to  the  double 
longitude  of  the  preceding  course  plus  the  longitude  differ- 
ence of  that  course  plus  the  longitude  difference  of  the 
course  itself. 

When  the  longitude  difference  is  negative,  or  a  westing,  it  is 
used  with  the  minus  sign  and  hence  subtracted  instead  of 
added.  If  the  meridian  is  drawn  through  the  most  westerly 
corner  of  the  field,  as  in  Fig.  9,  all  the  double  longitudes  are 
positive.  As  a  check  on  the  work  the  double  longitude  of  the 
last  course  will  be  found  equal  to  its  westing  ;  thus  the  double 
longitude  of  EA  is  eE. 

The  following  steps  in  the  computation  of  the  area  of  a  po- 
lygonal field  may  now  be  enumerated  : 

1st.  Measure  the  length  of  each  side  or  course  and  each  of 
the  interior  angles  ;  these  constitute  the  field  notes.  Also 
measure  the  azimuth  of  one  of  the  courses,  or  if  this  is  not 
measured  assume  any  value  for  this  azimuth. 

2d.  Compute  the  azimuth  of  each  of  the  other  courses  (Art.  2). 

3d.  Compute  the  latitude  difference  and  the  longitude  differ- 
ence for  each  course  (Art.  3). 

4th.  Compute  the  double  longitude  for  each  course. 

5th.  Multiply  each  double  longitude  by  its  latitude  differ- 
ence ;  call  the  positive  products  north  areas,  and  the  negative 
products  south  areas. 

6th.  Take  the  sum  of  the  south  areas  and  the  sum  of  the  north 
areas  ;  one  half  of  their  difference  will  be  the  area  of  the  field. 

In  Art.  6  a  numerical  example  will  be  given  illustrating  the 
computations  in  full. 

Prob.  5.  A  triangle  ABC  has  sides  with  the  following 
lengths  and  azimuths : 

AB,        I  =  312.0  feet,       Z  =    45  degrees. 
BC,         I  =  540.4  feet,       Z  =  135  degrees. 
CA,        I  =  624.0  feet,       Z  =  285  degrees. 
Compute  the  latitude  differences,  the  longitude  differences 
and  the  double  longitudes  for  each  course. 


20 


FUNDAMENTAL   PRINCIPLES. 


ART.  6.    COMPUTATION  OF  AREAS. 

The  following  are  the  lengths  of  the  sides  and  the  interioi 
angles  of  a  polygon  as  measured  in  surveying  a  field: 

AB  =  816.5  feet,  A  =    58°  14' 

BC  =  510.0  feet,  B  -  120  00 

CD  =  204.0  feet,  G  =  125  00 

DE  =  102.1  feet,  D  =  200  00 

EF  =  612.0  feet,  E  =    83  34 

FA  =  714.7  feet,  F  =  133  12 

No  azimuth  was  taken  in  the  field,  and  hence  for  the  pur- 
pose of  computing  the  area  the  meridian  is  assumed  to  pass 
through  AB,  so  that  the  azimuth 
of  AB  is  0°  00'. 

The  first  step  is  to  find  the  azi- 
muths of  the  other  sides  by  the 
method  of  Art.  3.  In  general  the 
azimuth  of  any  course  is  equal  to 
that  of  the  preceding  course,  plus 
180  degrees,  minus  the  interior  an- 
gle between  the  two  courses.  Thus 
the  azimuth  of  BC  is  0°  +  180°  - 
120°  =  60°;  the  azimuth  of  CD  is 
60° -|-  180°  -  125°  =  115°,  and  so  on. 
As  a  check  on  the  work  the  azimuth  of  AB  computed  from 
that  of  FA,  should  be  found  to  be  0°  00'. 

The  latitude  and  longitude  differences  of  the  courses  are  next 
computed  as  follows,  by  Art.  3  : 

Lat.  Diff.  AB  =  816.5  cos  0°  00'  =  +  816.50 
Lat.  Diff.  BC  =  510.0  cos  60°  00'  =  +  255.00 
Lat.  Diff.  CD  =  204.0  cos  115°  00'  =  -  86.21 
Long.Diff.  AB  =  816.5  sin  0°  00'  =  0.00 

Long.Diff.  BC=  510.0  sin  60°  00'  =  +  441.67 
Long.Diff.  EF=  612.0  sin  191°  26'  =  -  121.32 

In  like  manner  all  the  latitude  and  longitude  differences  are 
computed  and  the  results  are  tabulated,  the  positive  latitude 
differences  being  northings  and  the  negative  ones  sou  things, 


FIG.  10. 


COMPUTATION   OF  AREAS. 


while  the  positive  longitude  differences  are  eastings,  and  the 
negative  ones  westings. 


Courses. 

Lengths, 
feet. 

Azimuths. 

Lat.    Differences. 

Long.  Differences. 

North- 
ings. 

South- 
ings. 

Eastings. 

West- 
ings. 

AB 
BO 
CD 
DE 
EF 
FA 

816.5 
510.0 
204.0 
102.1 
612.0 
714.7 

0°  00' 
60  00 
115  00 
95  00 
191  26 
238  14 

816.50 
255.00 

86.21 
8.89 
599.85 
876.26 

0.00 
441.67 

184.89 
101.71 

0.00 

121.32 

607.65 

Totals  
Errors  

1071.50 

10T1.22 

728.27 

723.97 

0.28 

0.70 

Since  the  survey  was  made  by  a  circuit  from  A  back  to  A  it 
is  evident  that  the  sum  of  the  northings  should  equal  the  sum 
of  the  southings  ;  also  the  sum  of  the  eastings  should  equal 
the  sum  of  the  westings.  In  practice  this  is  rarely  attained, 
but  there  is  an  error,  called  the  error  of  closure,  which  should 
be  adjusted  before  the  double  longitudes  are  computed.  In 
this  case  the  significance  of  the  errors,  0.28  feet  in  latitude  and 
0.70  feet  in  longitude  is  that,  if  starting  from  A,  the  corners 
were  to  be  accurately  located  from  the  above  data,  the  end  A' 
of  the  line  FA'  would  fall  0.28  feet  to  the  north  of  A  and  0.70 
feet  west  of  it. 

The  error  of  closure  is  caused  by  errors  in  the  measurement 
of  the  lines,  or  in  observing  the  angles,  or  in  both.  However, 
if  the  sum  of  the  interior  angles  of  the  polygon  equals  180° 
into  the  number  of  sides  minus  360°,  the  probability  is  that 
the  error  of  closure  is  mostly  due  to  the  linear  measures.  As 
the  error  in  measuring  a  line  increases  with  its  length,  the 
error  in  latitude  should  be  distributed  among  all  the  latitude 
differences  in  proportion  to  their  lengths,  one  half  of  it  being 
applied  to  the  northings  and  one  half  to  the  southings.  The 
error  in  longitude  is  treated  in  the  same  way.  Thus  in  this 
case  the  errors  per  foot  in  latitude  and  longitude  are 


0.14 


=  0.000135, 


0.35 


=  0.000481, 


1071  ~  728 

and  the  adjusted  latitude  and  longitude  differences  are  found 
as  follows: 


FUNDAMENTAL   PRINCIPLES. 


Northing  AB  =  816.50  -  0.000135  X  816  =  816.39 
Southing  CD  =  86.21  +  0.000135  X  86  =  86.22 
Easting  BC  =  441.67  +  0.000481  X  442  =  441.88 
Westing  EF  -  121.32  -  0.000481  X  121  =  121.26 
and  their  values  are  inserted  in  the  table  given  below. 

The  double  longitudes  of  the  courses  are  next  computed. 
For  the  course  AB,  the  double  longitude  is  its  departure  0.00, 
for  the  second  course  BG  it  is  441.9,  for  CD  it  is  451.9  + 
441.9  +  185.0  =  1068.8,  and  so  on.  As  a  check  on  the  work 
the  double  longitude  of  the  last  course  will  be  found  equal  to 
its  westing.  The  fifth  column  of  the  table  gives  all  the 
double  longitudes. 


Courses. 

Adjusted 
Lat.  Differences 

Adjusted 
Long.  Differences 

Double 
Longi- 
tudes. 

Double  Areas. 

N. 

S. 

E. 

W. 

North. 

South. 

AB 
BO 
CD 
DE 
EF 
FA 

816.4 
255.0 

86.2 
8.9 
600.0 
376.3 

0.0 
441.9 
185,0 
101.8 

0.0 

121.3 
607.4 

0.0 
441.9 
1068.8 
1355.6 
1336.1 
607.4 

0 

112685 

92131 
12  065 
801660 
228565 

1071.4 

1071.4 

728.7 

728.7 

112685 

1  134  421 

The  fifth  step  is  to  multiply  the  double  longitude  of  each 
course  by  its  adjusted  latitude  difference,  and  to  place  the 
products  in  the  columns  of  double  areas.  Lastly  each  of  these 
columns  is  added,  and  then  the  double  area  of  the  field  is 

1 134  421  -  112  685  =  1 021 736  square  feet, 
and  accordingly  the  required  area  is  510  868  square  feet, 
which  is  equal  to  11  acres,  116  rods,  and  127  square  feet. 

This  result  can  be  verified  by  making  another  computation  in 
which  the  meridian  is  assumed  to  pass  through  some  other 
side,  as  BC.  Then  the  azimuth  of  BC  will  be  0°00',  that  of  CD 
will  be  55°  00'  and  so  on.  A  new  set  of  latitude  and  longitude 
projections  is  computed  and  these  are  adjusted  in  the  man- 
ner explained.  The  double  longitudes  of  the  courses  are  then 
found  and  each  is  multiplied  by  its  corresponding  northing 
or  southing.  Lastly  one  half  of  the  difference  of  these  pro- 
ducts will  give  the  area  in  square  feet,  which  should  closely 
agree  with  the  result  found  above. 


DIVISION   OF  LAND.  23 

Prob.  6.  Compute  the  area  of  the  above  field  taking  the  azi- 
muth of  BG  as  0°  00';  also  taking  the  azimuth  of  EF  as  0°  00'; 
also  taking  the  azimuth  of  AB  as  90°  00'. 


ART.  7.    DIVISION  OF  LAND. 

An  infinite  number  of  problems  may  arise  in  the  division  of 
a  field.  The  simpler  ones  will  be  readily  solved  by  the  use  of 
the  principles  of  geometry.  The  more  difficult  ones  can  be 
solved  after  a  complete  survey  of  the  field  and  the  computation 
of  its  area  has  been  made. 

The  first  problem  to  be  considered  is  that  of  dividing  a  field 
into  two  given  parts  by  a  line  starting  from  a  given  point.  As 
an  example  let  the  field  whose  area  was 
computed  in  Art.  6  be  taken,  and  let  it 
be  required  to  draw  from  the  point  Dt 
a  line  DP  so  that  the  area  BCDP  shall 
be  5  acres,  or  217  800  square  feet.  The 
solution  of  the  problem  involves  the 
determination  of  the  distance  AP  or 
BP,  and  of  the  length  and  azimuth  of 
the  dividing  line  DP.  (Fig.  11.) 

Let  a  line  be  drawn  from  D  to  the 
corner  A,  and  suppose  that  the  area 
ABGDA  can  be  found.  Then  the  area 
of  the  triangle  APDA  is  known,  as  this  is  equal  to  ABGDA 
minus  5  acres.  The  longitude  dD  of  the  point  D  is  also 
known.  Hence  the  length  of  AP  is 

.„       2  area  of  APDA 
~~ ,  y-. ; 

and  then  PB  =  AB  —  AP.     The  length  and  azimuth  of  DP 
are  finally  computed  from  the  right  triangle  of  dDP. 

To  perform  the  computations  for  finding  the  area  ABGDAt 
the  adjusted  latitude  and  longitude  differences  of  the  courses 
from  A  to  D  are  to  be  taken  from  Art.  6  and  inserted  in  the 
new  table  given  below.  The  latitude  difference  of  the  course 
DA  is  then  found  from  the  principle  that  the  sum  of  the  north- 
ings must  equal  the  sum  of  the  southings,  and  the  longitude 


FIG.  11. 


FUNDAMENTAL  PRINCIPLES. 


Courses. 

Latitude 
Differences. 

Longitude 
Differences. 

Double 
Longi- 
tudes. 

Double  Areas. 

N. 

S. 

E. 

W. 

North. 

South. 

AB 
BO 
CD 
DA 

816.4 
255.0 

86.2 
(985.2) 

0.0 
441.9 
185.0 

0.0 
(626.9) 

0.0 
441.9 

1068.8 
626.9 

0 

112685 

92131 
617  622 

1071.4 

1071.4 

626.9 

626.9 

112685 

709753 

difference  of  DA  is  supplied  in  like  manner.  Completing  then 
the  computations,  the  area  A  BCD  A  is  found  to  be  298  534 
square  feet.  The  area  of  the  triangle  A  DP  is  this  quantity 
minus  217  800  square  feet,  and  the  distance  AP  is 

2  X  80734 


626.9 


=  257.6  feet; 


whence  PB  is  558.8  feet,  and  hence  the  point  P  can  be  located 
from  either  A  or  B.     The  azimuth  of  PD  is  determined  thus, 


626.9 


255.0  -  86.2 


from  which  the  angle  dPD  is  found  to  be  40°  45'  nearly,  which 
is  the  azimuth  of  PD.     Lastly  the  length  of  PD  is 

dD 


PD  = 


sin  Z 


=  960.4  feet, 


and  thus  the  field  is  divided  by  the  line 
DP  so  that  the  area  BCDP  is  5  acres. 

A  second  problem  is  that  of  dividing  a 
field  into  two  parts  by  a  line  having  a 
given  direction.  For  example,  let  it  be 
required  to  divide  the  field  ABCDEF  into 
two  parts  by  a  line  PQ  so  that  the  azimuth 
of  PQ  shall  be  45  degrees  and  the  area 
PBCDQ  shall  be  5  acres  (Fig.  12).  First, 
the  computation  of  the  entire  field  is  to  be  made  as  in  Art.  6. 
Secondly,  a  line  DM  is  drawn  from  the  corner  D,  parallel  to 
QP,  and  by  the  method  above  described  the  area  MBCDM  is 
found  to  be  186224  square  feet  and  the  length  of  DM  to  be 


FIG.  12. 


INACCESSIBLE   DISTANCES.  25 

886.6  feet.  The  area  of  the  trapezoid  PMDQ  is  hence  to  be 
31576  square  feet.  Let  x  be  the  altitude  of  this  trapezoid; 
its  area  is  $(MD  +  PQ)x.  But  PQ  =  MD  +  x  cot  QPM + 
x  cot  DQP.  Hence 

i(2JfD  +  a?  cot  QPM  +  x  cot  D§P)z  =  31 576. 
Since  QPM  =  45°  and  DQP  =  50°,  this  reduces  to 

a;2  +  964.2*  =  34338, 

from  which  x  is  found  to  be  34.4  feet.     Then 
A/P  =  34.4/sin  45°  =  48.6  feet, 
DQ  =  34.4/sin  50° =45-0  feet, 
PQ  =  8S6.6  +  34.4-1.8391  =  949.8  feet, 

and  lastly  the  distance  AP  is  found  to  be  310.1  feet.  Thus 
P  and  Q  are  located  so  that  PQ  has  the  azimuth  45°,  and  the 
area  PBODQP  is  5  acres.  This  computation  may  now  be 
checked  by  computing  the  area  of  APQEFA,  which  should 
be  found  to  be  293  068  square  feet. 

Prob.  7.  Divide  the  field  ABCDEFA  into  two  equal  parts 
by  a  line  PQ  drawn  from  the  middle  point  of  AB.  Also  divide 
it  into  two  equal  parts  by  a  line  PQ  drawn  perpendicular  to 
the  side  AB. 

ART.  8.    INACCESSIBLE  DISTANCES. 

A  common  problem  in  surveying  is  to  find  the  horizontal  dis» 
tance  between  two  points  when  one  or  both  of  them  ai-e  in- 
accessible. This  can  be  solved  in 
many  ways  by  the  application  of  the 
principles  of  geometry  and  trigo- 
nometry. 

In  Fig.  13  let  A  be  an  accessible 
point  and  X  &n  inaccessible  point  on 
the  other  side  of  a  river.  It  is  re- 
quired to  find  the  distance  AX  by  Br 
means  of  the  chain  alone.  Place  a 
point  D  at  any  convenient  position 
in  the  prolongation  of  XA,  lay  off  a 
distance  AB,  make  BO  equal  to  AD,  FIG.  13. 

and  DC  equal  to  AB,  thus  forming  a  parallelogram  ABCD. 


FDTfTDAMZHTAI  PBISTCIPLES. 


Mark  a  point  E  where  XG  cuts  AB,  measure  AE,  EB,  and  Btt 
Then  from  the  similar  triangles  CB E  and  . 

AX  = 


by  which  the  required  distance  can  be  computed. 

By  the  use  of  an  instrument  for  measuring  angles  the  field 
operations  become  much  simpler,  and  indeed  the  method  by  the 
chain  is  often  impracticable  when  AX  is  a  long  line.  Let  (in 
Fig.  13)  a  line  AEbe  measured,  and  also  the  two  angles  A  and 
E\  then  the  angle  X  is  180°  —  A  —  E,  and 


which  is  the  required  distance.  The  base  line  AE  should 
usually  be  nearly  as  long  as  the  distance  AX  in  order  to  secure 
the  most  accurate  result,  and  it  is  also  well  that  the  angles  A 
and  E  should  be  approximately  equal. 

The  problem  of  two  inaccessi- 
ble points  is  illustrated  in  Fig.  14. 
Here  the  distance  XYis  required, 
and  for  this  purpose  a  base  line 
AB  is  measured  in  a  convenient 
location,  and  as  nearly  parallel 
to  XT  as  practicable.  At  A  the 
angles  XAB  and  TAB  are  ob- 
served, and  at  B  the  angles  AB  Y 
and  ABX.  Then  -in  the  triangle 

XAB, 

ABX 


FIG.  14. 


BXA  =  180°  -  XAB  -  ABX,     AX  =  AB 
Also  in  the  triangle  TAB, 

BYA  =  180°-  YAB  -  ABY,      AY  =  AB 


sin  BXA' 


sin  ABY 


sin  BYA' 

Thus  AX  and  AY  are  known,  and  the  angle  included  be- 
tween them  is  XA  Y  =  XAB  —  YAB  ;  then  in  the  triangle 
XA  Y  the  angles  at  X  and  Y  can  be  found  by  either  of  the 
methods  of  Art.  1,  and  lastly  the  distance  XY.  As  a  check  on 
the  work  the  sides  BX  and  B  Y  may  be  computed,  and  the 
distance  XY  be  again  found  from  the  triangle  XB  Y. 


ELEVATIONS   AND    HEIGHTS. 


27 


For  example,  let  it  be  required  to  find  the  horizontal  distance 
between  two  spires  X  and  Y.  The  base  AB  is  laid  off  406.2 
feet  in  length,  and  the  measured  angles  are  XAB  =  83°  47', 
TAB  =  42°  32',  ABT  =  76°*52',  and  ABX  =  36?  20'.  Then 
the  side  BY  is  found  to  be  315.2  feet,  BX  to  be  466.83  feet, 
and  their  included  angle  is  40°  32'.  The  angles  BYX  and 
YXB  are  next  found  to  be  97°  26'  and  42°  02',  respectively 
Lastly,  the  required  distance  XYis  306.0  feet. 

Prob.  8.  In  order  to  find  the  horizontal  distance  between  the 
tops  of  two  peaks  a  base  line  5000  feet  long  was  laid  off.  At 
one  end  of  the  line  the  angles  between  the  base  and  the  peaks 
were  120°  and  50°,  at  the  other  end  of  the  line  they  were  95° 
and  40°.  Find  the  distance  between  the  peaks,  and  check  the 
computation. 

ART.  9.     ELEVATIONS  AND  HEIGHTS. 

The  difference  in  level  between  two  points  on  the  ground 
which  are  accessible  is  usually  found  by  means  of  a  leveling 
instrument  and  a  graduated  rod.  The  level  is  placed  in  a 
horizontal  plane  by  means  of  its  bubble,  and  horizontal  sights 
are  taken  upon  the  rod  held  vertical  at  each  of  the  points. 
Thus  in  the  figure  to  find  the  difference  in  level  between  A  and 


FIQ.  15. 

B  the  level  is  placed  between  them;  the  rod  is  first  held  at  A, 
and  the  distance  a  is  read  between  the  foot  of  the  rod  and  the 
point  where  the  horizontal  line  through  the  level  cuts  it,  the 
rod  is  next  moved  to  B  and  the  distance  &a  is  there  read;  then 
the  difference  in  level  of  A  and  B,  or  the  elevation  of  A  above 
Bt  is  61  —  a.  When  the  difference  of  level  between  two  points 
A  and  C  is  greater  than  the  length  of  the  rod,  the  level  is  set 
up  twice,  as  shown  in  Fig.  15;  then  the  difference  of  level  be- 
tween A  and  G  is  61  —  a  -|-  c  —  62.  This  process  may  be  con- 


28 


FUNDAMENTAL   PRINCIPLES. 


tinned  as  many  times  as  necessary,  and  the  difference  in  level 
between  the  initial  and  final  points  is  then  the  sum  of  the 
forward  readings  minus  the  sum  of  the  backward  readings. 

The_elevation  of  a  point  is  its  height  above  sea  level  or  above 
some  datum  plane.  In  running  levels  it  is  customary  to  start, 
from  some  point,  called  a  bench-mark,  whose  elevation  is 
known.  Thus,  in  Fig.  15,  let  the  point  A  be  a  bench-mark 
whose  elevation  is  328.72  feet,  and  let  the  reading  a  be  0.93 
feet,  61  be  10.84  feet,  62  be  1.03  feet,  and  c  be  11.47  feet.  Then 
the  elevation  of  B  is  318.81  feet  and  the  elevation  of  G  is 
308.37  feet. 

The  height  of  an  inaccessible  point  is  usually  found  by  the 
help  of  vertical  angles  together  with  a  measured  base  and 

certain  horizontal 
angles.  Let  it  be 
required  to  find 
the  height  of  the 
top  of  the  flag^ 
pole  X  above  the 
point  Y  at  the 
base  of  the  build- 
ing. In  any  con- 
venient position 
let  a  horizontal 
base  AB  be  meas- 
ured, also  let  the 
horizontal  angles 
CBA  and  BAG  be  measured  where  G  is  a  point  vertically 
below  X  and  at  the  same  elevation  as  A  ;  in  reality  no  point 
G  is  established,  but  these  angles  are  measured  by  pointing  the 
instrument  at  X,  the  angle  GBA  being  the  horizontal  projec- 
tion of  the  angle  XBA.  The  horizontal  angles  DBA  and 
BAD  are  likewise  measured  where  D  is  a  point  vertically 
above  Y.  At  A  the  vertical  angles  XAG  and  YAD  are  also 
measured. 

In  the  triangle  ABG  two  angles  and  one  side  are  now  known, 
and  from  these  the  horizontal  line  AG  is  computed.  Then 
in  the  right  triangle  AGX  the  side  AG  and  the  vertical  angle  at 


EKRORS  OF  MEASUREMENTS.  29 

A  are  known,  and  from  these  the  vertical  height  XC  is  com- 
puted.  Again,  in  the  triangle  ABD  two  angles  and  one  side 
are  known,  from  which  the  horizontal  side  AD  is  found  ;  then 
in  the  right  triangle  AD  Tike  vertical  side  DY  is  computed 
from  AD  and  the  vertical  angle  at  A.  Finally,  the  required 
height  XT  is  the  sum  of  Xtfand  YD. 

As  an  example,  let  the  base  AB  he  314.62  feet,  CBA  =  40°  17', 
DBA  =  38°  22',  BAG  =  48°  40',  BAD  -  46°  57',  while  the  ver- 
tical angles  at  A  are  XAG  —  37°  18'  and  TAD  =  5°  08'.  Then 
the  side  AC  is 

AC=  314.62^-|^4|-!  =  203.46  feet, 
sin  91   03 

and  in  like  manner  AD  is  found  to  be  195.80  feet.     Then 

XC  =  AC  tan  37°  18'  =  154.99  feet ; 
TO  =  AD  tan    5°  08'=    17.59    " 

aad,  lastly,  the  height  XY  is  154.99  +  17.59  =  172.6  feet,  the 
second  decimal  being  omitted,  as  it  is  probably  inaccurate. 

In  case  that  Y  is  a  point  on  the  building  above  the  level  of 
the  instrument  at  A,  as  may  often  happen,  then  XY  is  the 
difference  of  XC  and  YD.  In  order  to  check  the  work  vertical 
angles  may  also  be  observed  at  B. 

Prob.  9.  In  order  to  find  the  difference  in  height  of  two 
peaks,  Jbfand  N,  a  base-line  AB  was  laid  off  5000  feet  long, 
and  the  horizontal  angles  BAM  =  120°  30',  BAN  =  49°  15', 
ABM  =  40°  35',  ABN  =  95°  07',  were  read.  At  A  the  angle 
of  elevation  of  M  was  17°  19',  and  the  angle  of  elevation  of  N 
was  18°  45'.  Compute  the  difference  in  height  of  the  two 
peaks. 

ART.  10.    ERRORS  OF  MEASUREMENTS. 

All  measurements  are  subject  to  errors  which  may  be  divided 
into  two  classes,  systematic  or  constant  errors,  and  accidental 
errors.  Systematic  errors  are  those  that  always  have  the  same 
value  under  the  same  circumstances,  being  due  to  known 
causes  ;  for  example,  if  a  100-foot  chain  be  one  foot  too  long, 
all  measurements  made  with  it  will  be  one  per  cent  too  short. 
Accidental  errors  are  those  that  are  equally  likely  to  render  the 


30  FUNDAMENTAL  PEINCIPLES. 

measurement  larger  or  smaller  than  the  true  value,  being  due 
to  the  combination  of  many  unknown  causes;  for  instance, 
variations  in  wind,  imperfection  of  eyesight,  and  other  similar 
causes  render  a  measurement  too  great  or  too  small. 

Systematic  or  constant  errors  can  be  removed  from  measure- 
ments, when  their  causes  are  understood,  either  by  a  proper 
method  of  observing  or  by  applying  proper  corrections  to  the 
numerical  results.  Methods  of  doing  this  for  both  linear  and 
angular  measures  will  be  given  in  the  following  chapters. 

After  all  the  systematic  errors  are  removed  the  numerical 
results  are  still  affected  by  the  accidental  errors.  As  these  are 
equally  likely  to  increase  or  decrease  the  true  value  of  the 
quantity  they  tend  to  balance  one  another,  and  hence  if  only 
one  measurement  be  made  it  must  be  accepted  as  the  most 
probable  value.  For  instance,  if  one  measurement  of  a  line 
gives  618.5  feet,  after  the  systematic  errors  are  removed,  that 
value  must  be  taken  as  representing  the  true  value. 

When  several  measurements  of  a  line  are  made  under  the 
same  conditions  each  has  the  same  degree  of  probability,  and 
hence  their  arithmetical  mean  is  to  be  taken  as  the  most  prob- 
able value  ;  for  example,  if  three  measures  of  a  line,  made  in 
the  same  manner,  gives  618.5,  619.1,  and  618.9  feet,  there  is 
no  reason  for  preferring  one  to  the  other,  and  hence  one  third 
of  their  sum,  or  618.83  feet,  is  to  be  taken  as  the  most  probable 
length. 

If  the  three  angles  of  a  triangle  are  measured  with  equal 
care  their  sum  should  be  180  degrees.  If  this  is  not  the  case 
the  results  are  to  be  adjusted  by  applying  one-third  of  the 
error  to  each  of  the  measured  angles.  So  with  a  polygon  of 
n  sides,  when  the  n  interior  angles  are  measured,  their  sum 
should  equal  ISOn  —  360  degrees,  and  if  this  is  not  the  case 
one-nth  of  the  error  should  be  applied  to  each  of  the  measured 
values  in  order  that  their  sum  may  equal  the  theoretic  amount. 

When  the  sides  and  angles  of  a  field  are  measured  the  sum 
of  the  northings  should  equal  the  sum  of  the  southings,  and 
also  the  sum  of  the  westings  should  equal  the  sum  of  the  east- 
ings. Owing  to  errors  in  measurement  these  conditions  will 


ERRORS   OF   MEASUREMENT.  31 

rarely  occur,  and  hence  an  adjustment  must  be  made,  as  ex- 
plained in  Art.  6,  to  remove  the  accidental  errors. 

When  three  angles  AOB,  BOG,  AOO  are  measured  at  a 
station  0  with  equal  care,  the  sum  of 
AOB  and  BOG  should  equal  AOG.  If 
this  is  not  the  case  an  adjustment  must 
be  made  by  applying  one-third  of  the 
error  to  each  angle.  For  example,  let 
the  measured  values  be  AOB  =  32°  16', 
50(7=55°  43',  and  AOG  =  87°  57';  FlG<  17' 

then  the  adjusted  values  are  AOB  =  32°  15'  20",  50(7  =  55' 
42'  20",  and  AOG  =  87°  57'  40",  which  exactly  satisfy  the  the- 
oretic condition.  It  is  always  advantageous  to  measure  the 
three  angles  even  if  only  two  are  required,  as  thus  a  check  is 
furnished  on  the  work  and  opportunity  is  offered  to  eliminate 
the  accidental  errors  of  the  measurements. 

The  young  surveyor  should  always  bear  m  mind  that  the 
results  of  his  measurements  in  the  field  are  not  the  true  values 
of  the  quantities  which  they  represent,  but  only  approximate 
representations  of  the  true  values.  He  should  seek  to  secure 
the  greatest  degree  of  precision  consistent  with  the  tools  em. 
ployed  and  the  end  in  view.  A  large  part  of  the  land  surveys 
in  the  United  States  has  been  made  by  rough  and  imperfect 
methods,  but  the  time  has  now  come  when  precision  is  de- 
manded. Hence  care  must  be  taken  to  make  sufficient  meas- 
urements so  that  the  work  can  be  checked,  to  remove  all  sys- 
tematic sources  of  error,  and  finally  to  adjust  the  results  when 
possible  so  that  the  accidental  errors  may  be  largely  eliminated. 
In  precise  triangulation  work  the  adjustment  of  measurements 
is  especially  important,  and  the  principles  and  methods  for 
doing  this  constitute  a  branch  of  science  known  as  the  method 
of  least  squares. 

Prob.  10.  At  a  point  0  four  angles  are  measured  as  fol 
lows  :  AOB  =  35°  07',  BOG  =  60°  43',  COD  =  22°  Ol': 
AOD  =  117°  53'.  Find  their  adjusted  values. 


32  LAND  SURVEYING. 


CHAPTER  II. 
LAND  SURVEYING. 

ART.  11.    CHAINS  AND  TAPES. 

THE  chains  used  in  land  surveying  are  made  of  steel  wire 
and  have  the  joints  brazed  to  prevent  opening.  Iron  chains 
are  seldom  used,  being  heavier  and  in  every  way  inferior  to 
those  made  of  steel.  At  intervals  of  10  links  brass  tags  are 
fastened,  having  one,  two,  three,  or  four  points,  indicating 
distances  of  ten,  twenty,  thirty,  or  forty  links  from  either  end; 
the  middle  of  the  chain  is  marked  by  a  round  tag.  The  chain 
is  provided,  at  either  end,  with  brass  handles  fastened  to  it  by 
a  nut  and  screw  by  which  the  length  may  be  changed  a  small 
amount.  The  length  of  the  chain  includes  the  handles.  In 
using  the  chain  care  must  be  taken  to  observe  whether  the  dis- 
tance is  greater  or  less  than  half  a  chain,  as  forty  links  and 
sixty  links  are  marked  alike,  and  thirty  links  from  seventy 
links,  as  also  twenty  links  from  eighty  links,  must  be  carefully 
distinguished. 

The  chain  is  folded  by  bringing  the  49th  and  51st  links  to- 
gether, the  48th  and  52d  together,  and  so  on  until  the  ends  ari 
reached,  folding  links  equidistant  from  the  middle  together. 
To  unfold  the  chain,  hold  both  handles  in  the  left  hand  and 
with  the  right  hand  throw  it  horizontally  far  enough  so  that  it 
will  become  taut  before  it  falls. 

The  chain  possesses  some  advantages  over  the  tape  on  account 
of  its  weight  and  strength,  and  because  it  can  be  more  easily 
repaired.  In  chaining  through  brush  the  weight  of  the  chain 
is  serviceable  in  swinging  it  over  the  bushes  and  in  making  it 
straight  and  horizontal.  If  the  chain  is  broken,  a  new  link 
may  be  put  in  by  the  surveyor. 

Steel  tapes  are  made  in  various  lengths  up  to  500  feet;  those 
having  lengths  of  50  feet  or  100  feet  are  generally  used  in 
land  surveying.  The  best  tapes  of  these  lengths  are  about  0.4 
inches  wide  and,  perhaps,  0.005  inches  thick:  they  are  grada- 


CHAIHS   AND   TAPES.  33 

ated  throughout  tlie  entire  length  into hundredths of  afoot,  and 
often  the  reverse  side  is  divided  *nto  rods  and  links.  These 
tapes  are  easily  broken,  and  are  only  used  where  the  value  of 
the  land  warrants  very  careful  measurements;  they  rust  easily 
and  should  be  wiped  dry  after  using,  and  all  small  spots  of  rust 
removed  with  kerosene. 

Tapes  used  in  common  land  surveying  are  narrower  and 
thicker  than  those  described  above;  the  first  foot  from  either 
end  is  divided  into  tenths,  the  first  and  last  five  foot  spaces  are 
divided  into  feet,  and  the  tape  throughout  is  marked  every  five 
feet.  When  nickel-plated  these  tapes  require  much  less  at- 
tention to  keep  them  from  rusting  than  the  finer  grades.  In 
nearly  every  point  of  difference  between  such  a  tape  and  the 
best  chain  the  comparison  is  in  favor  of  the  tape  ;  one  great 
advantage  is  that  wear  does  not  increase  its  length  to  the  same 
degree  as  in  a  chain. 

Metallic  tapes,  so  called,  are  made  of  cloth,  and  have  strands 
of  fine  brass  wire  interwoven  longitudinally.  They  are  divided 
throughout  into  tenths  of  a  foot,  and  are  very  useful  in  making 
short  measurements  when  great  accuracy  is  not  required,  as  in 
finding  the  dimensions  of  buildings,  taking  offsets  to  locate 
paths,  brooks,  and  other  details  of  topography. 

To  use  the  tape  or  chain,  two  men  are  required,  called 
respectively  the  head  chainman  and  rear  chainman.  The  chain 
is  brought  into  the  line  and  made  level  with  the  rear  end  over 
the  first  point ;  the  head  chainman,  by  means  of  a  plumb-bob, 
finds  the  spot  directly  under  the  front  end  of  the  chain,  and 
marks  it  by  a  nail  or  iron  pin  made  for  the  purpose.  This 
operation  is  repeated  till  the  end  of  the  line  is  reached. 

If  pins  are  used  there  should  be  eleven  of  them.  The  head 
chainman  places  a  pin  at  the  front  end  of  the  chain,  and  this  is 
taken  up  by  the  rear  chainman  after  the  head  chainman  has 
placed  a  second  pin.  When  the  last  pin  is  in  the  ground  the 
rear  chainman  delivers  his  ten  pins  to  the  head  chainman  and 
the  work  is  continued.  Each  delivery,  which  is  generally 
called  a  tally,  thus  indicates  ten  chain  lengths. 

In  using  the  plumb-bob  with  the  chain,  it  is  best  to  stand 


34  LAKD   StJRYEYING. 

facing  across  the  line  to  be  measured  ;  the  string  is  held  against 
the  proper  point  on  the  chain  with  the  thumb  and  forefinger  ot 
the  right  hand,  and  the  left  hand,  pressing  against  them,  helpa 
in  stretching  the  chain.  The  head  chainman,  after  finding 
approximately  where  the  point  will  be,  should  carefully  clear 
away  all  leaves  and  grass,  and  prepare  a  smooth  place  on  the 
ground,  so  that  a  slight  touch  of  the  plumb-bob  may  be  suffi- 
cient to  mark  the  point. 

In  passing  along  the  line  the  rear  end  of  the  chain  is  allowed 
to  drag  along  the  ground,  and  just  before  it  reaches  the  pin 
the  head  chainman  is  notified  of  the  fact  by  some  preconcerted 
signal,  such  as  " chain"  or  "chain  out";  much  time  can  be 
saved  by  stopping  the  head  chainman  at  just  the  proper  time. 

On  steep  slopes  it  is  best  to  chain  down  hill.  When  the 
difference  in  elevation  of  the  ground  along  the  line  is  more 
than  six  or  seven  feet  in  a  hundred  feet,  the  head  chainman 
carries  his  end  of  the  chain  out  as  usual  and  puts  it  in  line;  he 
then  goes  back  to  a  place  which  is  not  more  than  six  feet  lower 
than  the  rear  end  of  the  chain  and  proceeds  in  usual  manner, 
except  that  a  part  instead  of  the  whole  of  the  chain  is  used, 
When  the  measurement  of  one  of  the  short  divisions  is  com- 
pleted, the  rear  chainman  holds  the  proper  division  over  the 
point  last  determined,  and  the  operation  is  repeated  till  the 
front  end  of  the  chain  is  reached.  It  is  unnecessary  to  record 
or  even  to  notice  the  lengths  of  the  divisions,  as  the  end  of  the 
chain  will  be  a  chain's  length  from  the  point  of  beginning. 
This  operation  is  called  "  breaking  the  chain." 

Instead  of  using  the  plumb-bob,  the  horizontal  distance  is 
often  found  in  accurate  work  by  measuring  along  the  surface 
of  the  ground,  and  afterwards  determining  the  difference  in 
height  of  points  between  which  the  measurements  were  taken. 
The  length  along  the  chain  then  represents  the  hyppthenuse  of 
a  right  triangle,  of  which  required  distance  is  another  side. 

A  chain  should  be  frequently  compared  with  a  standard  laid 
off  on  a  floor  or  pavement.  For  common  work  in  land  survey- 
ing, such  a  standard  may  be  laid  off  by  a  good  steel  tape  which 
has  not  been  used.  For  precise  work  in  cities  the  steel  tape 


THE   TRANSIT.  35 

itself  should  be  standardized,  which  can  be  done  by  the  depart- 
ment of  Weights  and  Measures  of  the  U.  S.  Coast  and  Geodetic 
Survey  at  Washington  (see  Art.  28). 

Many  surveyors  prefer  to  have  a  chain  a  little  longer  than 
the  standard  in  order  to  compensate  for  lack  of  level  and  for 
lateral  deviations.  In  good  work,  however,  these  sources  of 
error  should  be  avoided,  and  the  chain  should  agree  exactly 
with  the  standard.  If  a  chain  is  too  long  the  measured  length 
of  a  line  is  too  small;  thus,  if  the  length  824.5  feet  be  obtained 
by  a  hundred-foot  chain  which  is  0.14  feet  too  long,  the  true 
length  of  the  line  is  8.245  (100  +  0.14)  =  825.7  feet.  If  a  chain 
is  too  short  the  measured  length  is  too  large  ;  thus  if  the  length 
785.8  feet  be  obtained  by  a  chain  which  is  0.07  feet  too  short, 
the  true  length  of  the  line  is  7.858  (100  —  0.07)  =  785.25  feet. 

Prob.  11.  A  careless  surveyor  measured  a  field  with  a 
hundred-foot  chain,  and  computed  the  area  to  be  8  acres,  12 
rods,  146  square  feet.  It  was  afterwards  found  that  the  chain 
had  lost  one  link,  so  that  its  true  length  was  only  99  feet.  If 
the  computations  of  the  surveyor  were  correct,  what  is  the  true 
area  of  the  field. 

ART.  12.    THE  TRANSIT. 

The  surveyor's  transit  consists  primarily  of  two  parts ;  the 
first,  called  the  alidade,  determines  the  line  of  sight,  and  the 
second,  called  the  limb,  affords  means  of  determining  the 
angular  deviation  of  this  line  from  any  other.  The  alidade,  in- 
cluding the  telescope,  the  magnetic  needle  with  its  graduated 
circle  and  the  vernier,  is  attached  to  a  vertical  spindle,  and 
may  be  revolved  while  the  limb  remains  stationary.  The  hori- 
zontal circle  composing  the  limb  is  graduated  into  degrees,  and 
sometimes  into  thirty  minute  or  twenty  minute  spaces,  and 
numbered  from  zero  to  360  degrees  in  both  directions.  The 
limb  is  mounted  upon  a  hollow  cylindrical  annulus  which  sur- 
rounds the  spindle  of  the  alidade.  The  instrument  is  supported 
by  three  legs,  called  the  tripod,  which  are  fastened  together  at 
the  top  by  the  tripod  head. 

The  device  used  to  measure  fractional  amounts  of  the  divi- 
sions of  the  limb  is  called  a  vernier.  Verniers  are  used  either 


36  LAND   SURVEYING. 

on  straight  or  circular  scales,  the  former  being  employed  on 
level  rods  and  the  latter  on  transits.  In  Fig.  18  is  shown  a 
vernier  for  a  straight  scale,  where  the  length  of  the  vernier  is 
the  same  as  the  length  of  nine  spaces  of  the  limb.  The  vernier 
itself  is  divided  into  ten  equal  parts.  Let  a  be  the  length  of 


1       u.*-* 

Scale    or    Limb 
III                        III 

1 

f-6-J        III                I        1        1        1 

1                                            1                     Vernier 

\          , 

i       Scale    or  i  Limb 
f                        1                        III 

j 

II                     1           1           1           1           1           1 

1                    Vernier 

FIG.  18. 

one  space  on  the  limb,  and  b  the  length  of  one  space  on  the 
vernier.  On  a  level  rod  a  is  T^th  of  a  foot,  then  b  is  j^th  of 
Tfj)th  of  a  foot,  hence 

a—b  =  ^-s  —  y&fl  =  T^VTJ  f eet  5 

and  thus  the  space  between  the  first  division  of  the  limb  and 
the  first  division  of  the  vernier  in  Fig.  18  is  yj^  of  a  foot,  or 
one-tenth  of  a  space  of  the  limb. 

If  the  vernier  in  the  first  diagram  of  Fig.  18  is  moved  until 
its  first  division  coincides  with  the  first  division  of  the  limb  a 
distance  of  -faa  or  y^Vir  ^ee^  ^as  ^een  passed  over.  If  the  third 
divisions  coincide,  as  the  second  diagram,  the  vernier  has 
moved  a  distance  of  -f^a  or  T¥?ffir  feet.  Thus  in  moving  the  ver- 
nier fractional  parts  of  the  smallest  space  of  the  limb  are  read 
with  precision  by  noting  what  division  of  the  vernier  coincides 
with  a  divisior  of  the  limb. 

If  the  length  of  the  vernier  is  equal  to  19  spaces  of  the  limb 
and  it  is  divided  into  20  parts,  the  distance  a  —  b  will  be  one- 
twentieth  of  one  space  of  the  limb,  or  a  degree  of  precision 
twice  as  high  as  before.  Hence  a  general  rule  for  finding  the 
smallest  amount  indicated  by  the  vernier  is  this  :  Divide  the 
value  of  the  smallest  space  of  the  limb  by  the  number  of  spaces 
on  the  vernier. 

A  vernier  can  be  also  made  by  making  its  length  equal  to  11 


THE  TKANSIT. 


37 


spaces  of  the  limb  and  dividing  it  into  10  equal  parts,  or  by 
making  its  length  equal  to  21  spaces  of  the  limb  and  dividing 
it  into  20  parts.  Such  an  arrangement  is  called  a  retrograde 
vernier,  and  is  not  commonly  used. 

The  verniers  used  on  transits  are,  of  course,  circular  instead 
of  straight,  and  the  divisions  on  the  limb  are  degrees  and  frac- 
tions of  degrees  instead  of  feet,  but  the  principles  do  not  differ 
from  those  stated  above.  Such  verniers  are  usually  made 
double  for  convenience  in  reading  angles  in  either  direction. 
Such  a  vernier  is  shown  in  Fig.  19.  Here  it  is  seen  that  the 
zero  point  on  the  vernier,  in  moving  from  the  right  to  the  left, 
has  passed  the  point  a,  which  is  66°  30',  and  is  at  b.  By  using 


the  vernier  it  is  possible  to  measure  the  space  ab.  In  the 
figure  the  limb  is  divided  into  thirty  minute  spaces,  the  ver- 
nier is  of  the  same  length  as  twenty -nine  of  these  spaces,  and 
is  divided  into  thirty  spaces.  Hence  the  smallest  amount  in- 
dicated by  such  a  vernier  will  be  the  difference  between  the 
lengths  of  a  space  on  limb  and  on  the  vernier,  or  one  minute. 
By  referring  to  the  figure  it  is  seen  that  the  fourth  division  on 
the  vernier  to  the  left  of  zero  coincides  with  one  on  the  limb, 
hence  the  zero  point  has  moved  four  minutes  after  passing  the 
point  a,  and  the  reading  is  66°  30'  +  04'  or  66°  34'. 

In  using  the  double  vernier  the  beginner  may  be  in  some 
doubt  as  to  which  part  to  use.  This  can  be  guarded  against 
by  reading  that  side  which  is  farthest  away  from  zero  on  the 
limb,  in  the  direction  that  the  vernier  has  been  turned. 

The  precision  of  the  work  done  by  an  instrument  depends  as 
much  upon  the  care  taken  of  it  as  upon  its  original  excellence. 


38  LAND   SURVEYING. 

In  carrying  the  transit  to  and  from  work,  care  must  be  taken 
that  the  tripod  is  firmly  attached  ;  the  telescope  should  be 
turned  in  line  with  the  axis  of  the  instrument,  but  not  too 
rigidly  clamped  ;  the  cap  should  be  placed  over  the  objective 
and  the  needle  lifted  from  the  centre  pin.  The  instrument, 
while  being  carried,  is  held  on  the  shoulder  by  the  hand  just 
in  front  with  the  elbow  close  to  the  side  ;  in  this  way  there  is 
more  freedom  of  movement  and  the  least  liability  to  accident. 

In  setting  up  the  instrument  it  is,  in  most  cases,  better  to  put 
two  legs  down  hill  and  one  leg  up  hill.  The  instrument  is 
lifted  bodily  and  set,  as  nearly  as  may  be,  over  the  point,  with 
the  plates  parallel  and  horizontal.  In  bringing  the  transit  into 
exactly  the  required  position  it  is  only  necessary  to  remember 
that  the  plumb-bob  will  follow  the  direction  in  which  either 
leg  is  made  to  move — toward  it  or  away  from  it  according  as 
the  leg  is  carried  out  or  in.  It  is  not  well  to  force  the  tripcd 
feet  further  into  the  ground  than  is  necessary  for  rigidity  ; 
some  tripods  are  wisely  furnished  with  lugs  to  receive  the 
pressure  from  the  foot ;  thus  the  tripod  head  is  relieved  of 
much  unnecessary  strain. 

After  the  instrument  has  been  set  up  with  the  plumb-bob 
over  the  point,  the  next  step  is  to  level  the  plates.  The  in- 
strument  is  first  turned  so  that  the  bubble  tubes  are  parallel 
to  the  lines  through  the  two  opposite  leveling  screws  ;  it  is 
then  leveled  by  turning  the  screws  in  opposite  directions;  this 
will  be  accomplished  when  the  thumbs,  in  turning,  move 
either  toward  or  from  each  other.  The  bubble  will  be  seen  to 
move  in  the  direction  in  which  the  left  thumb  moves.  After 
all  the  leveling  screws  are  brought  to  a  bearing  on  the  plates 
by  turning  one  screw  in  each  pair,  they  should  only  be  turned 
in  pairs  and  in  opposite  directions;  in  this  way  the  bearing 
upon  the  plates  will  be  preserved  and  the  screws  and  plates 
will  not  become  strained. 

Suppose  the  transit  to  be  set  over  the  point  0  in  Fig.  17  and 
that  it  is  desired  to  measure  the  horizontal  angle  AOB.  The 
telescope  is  directed,  with  the  vernier  clamped,  toward  either 
of  the  points  B  or  A,  and  the  limb  clamped  ;  the  vernier  is 


THE    TRANSIT.  39 

then  read  and  undamped,  and  the  telescope  is  directed  toward 
the  other  point,  the  alidade  clamped,  and  the  vernier  read 
again.  It  is  evident  that,  as  the  vertical  plane  of  the  telescope 
and  the  vernier  are  relatively  immovable,  the  angular  distance 
passed  over  by  the  zero  point  on  the  vernier  and  by  the  plane 
of  the  telescope  are  the  same,  or  the  angle  AOB.  Hence,  to 
measure  an  angle,  readings  of  the  vernier  are  made  before  and 
after  the  angle  is  turned,  and  the  difference  is  taken.  In  or- 
dinary work  it  is  usual  to  set  the  vernier  at  zero  before  turning 
the  angle,  in  which  case  the  reading  after  the  second  sight  has 
been  taken  is  the  angle  itself. 

It  is  only  necessary  to  follow  the  above  directions  to  cor^ 
rectly  measure  any  angle,  but  the  operation  can  seldom  be  done 
by  a  beginner  so  that  no  errors  are  involved.  It  is  readily  seen 
that  the  accuracy  of  the  measurement  of  an  angle  depends 
upon  the  following  : 

The  adjustment  of  the  transit. 

Setting  the  instrument  over  the  exact  point  it  is  desired  to 
have  it  occupy. 

The  reading  of  the  vernier. 

The  bisection  of  the  points  toward  which  the  telescope  is 
directed. 

The  movement  of  the  alidade  due  to  defects  in  clamping. 

In  land  surveying  where  angles  are  only  read  to  the  nearest 
minute  these  errors  should  be  made  as  small  as  possible  by 
seeing  that  the  transit  is  in  adjustment,  that  it  is  set  over  the 
exact  centre  of  tl^e  station,  that  the  vernier  is  accurately  read, 
that  the  signals  sighted  upon  are  correctly  placed  and  truly 
bisected,  and  that  care  is  taken  in  using  the  clamps.  Direc- 
tions for  adjusting  a  transit  are  given  in  Art.  27,  but  a  beginner 
should  never  attempt  to  make  them  until  he  has  used  the  instru- 
ment sufficiently  to  become  thoroughly  acquainted  with  all 
the  manipulations. 

In  precise  work  where  angles  are  needed  to  fractions  of  a 
minute  the  last  three  sources  of  error  mentioned  above,  as  well 
as  some  others,  may  be  largely  eliminated  by  the  method  of 
repetitions  described  in  Art.  28.  In  land  surveying  repetitions 
are  unnecessary,  but  it  will  be  well  to  check  each  angle  by 


40  LAND   SURVEYING. 

measuring  also  its  explement.  Thus,  if  the  angle  AOB  is  read 
by  pointing  first  on  A  and  then  on  B,  let  the  angle  BOA  be  read 
by  pointing  first  on  B  and  then  on  A  ;  the  sum  of  the  two 
angles  should  be  360°  00'. 

An  engineer's  /transit  mainly  differs  from  a  surveyor's  transit 
in  having  a  vertical  arc  and  a  level  bubble  attached  to  the  tele- 
scope for  the  determination  of  heights  and  elevations.  Some 
engineers'  transits  have  verniers  reading  to  half-minutes,  while 
transits  for  triangulation  work  sometimes  read  to  twenty 
seconds  or  to  ten  seconds. 

Prob.  12.  If  the  limb  is  divided  into  20-minute  spaces,  show 
how  the  vernier  must  be  made  in  order  to  read  one  minute? 
in  order  to  read  20  seconds  ?  Give  diagrams  of  these  verniers. 

ART.  13.    THE  MAGNETIC  NEEDLE. 

Most  of  the  early  land  surveys  of  the  United  States  we^fc 
made  by  the  compass.  The  compass  is  an  instrument  like  tl\e 
surveyor's  transit,  but  without  graduated  limb  and  telescope  ; 
the  place  of  the  latter  is  supplied  by  vertical  sights,  whi)o 
angles  are  read  by  bearings  of  the  magnetic  needle.  All  the 
remarks  here  made  regarding  the  magnetic  needle  apply 
equally  to  the  compass  and  to  the  transit,  although  in  the  case 
of  the  transit  the  needle  is  used  less  than  the  graduated  limb 
and  vernier. 

The  compass  plate  is  usually  graduated  to  half-degrees  ;  the 
north  and  south  points,  lettered  2?  and  S,  are  marked  0°,  and 
the  graduation  runs  from  each  in  both  directions  to  the  east 
and  west  points  which  are  marked  90°.  The  letters  E  and  W 
are,  however,  on  the  west  and  east  sides  respectively,  of  the 
compass  plate,  in  order  that  the  direction  of  a  line  as  read  from 
the  end  of  the  needle  may  agree  with  its  actual  direction.  The 
direction  of  a  line  as  determined  by  the  needle  is  called  its 
magnetic  bearing.  The  bearing  is  expressed  by  two  of  the 
letters  JV,  Et  8,  or  W,  with  the  number  of  degrees  which  the 
line  varies  from  the  magnetic  meridian  ;  thus  ^35°  E,  which 
is  read  north  thirty-five  degrees  east,  means  a  line  whose  direc- 
tion is  thirty-five  degrees  east  of  north  ;  also  8  70°  FT  indicates 


THE    MAGNETIC   XEEDLE. 


41 


W  — 


a  line  whose  magnetic  direction  is  seventy  degrees  west  of 
south. 

When  the  bearings  of  several  lines  are  taken  at  the  same 
point  the  angles  between  them  are  known.  For  example,  let 
the  bearing  of  AC  be  NS^°  E,  and 
that  of  AD  be  ^46°  E,  then  the  angle 
CAD  is  374  degrees.  Also  if  the  bear- 
ing of  AF  be  8  52fc°  E,  then  the  angle 
DAF  is  81i  degrees.  The  student 
should  deduce  his  own  rule  for  find- 
ing the  angle  from  the  bearings  by 
drawing  figures  for  a  few  special 
cases. 

When  the  bearings  of  several 
courses  are  given  the  angles  between  them  are  also  known. 
Thus,  in  Fig.  21  let  the  bearing  of  AB  be  ^42°  E,  and  that 
of  BC  be  8  29±°  E;  then  the  angle  ABC  is  71i°.  Here  it  is 
best  to  reverse  the  bearing  of  the  first  line,  and  thus  considei 
both  as  taken  at  the  point  B  where 
the  bearing  of  BA  is  8  42°  W. 

The  magnetic  needle  is,  at  the  best, 
a  rough  and  imperfect  tool  for  meas- 
uring angles  or  for  determining  the 
directions  of  lines.  The  bearings  can 
be  read  to  quarters  or  eighths  of  a 
degree,  but  owing  to  the  variations  to 
which  the  needle  is  subject,  a  line  will 
have  different  bearings  at  different 
times.  The  magnetic  meridian  at  most  places  deviates  from  the 
true  meridian,  and  the  angle  between  them  is  called  the  declina 
tion  of  the  needle.  On  the  Atlantic  coast  of  the  United  States 
the  declination  is  to  the  west  of  the  true  meridian,  while  on  the 
Pacific  coast  it  is  to  the  east,  but  its  amount  is  very  different  in 
different  places,  as  will  be  seen  from  the  isogonic  map  of  the 
United  States  for  1915  inserted  at  page  128  of  this  Handbook. 
An  isogonic  line  is  a  curve  passing  through  all  places  which 
have  the  same  magnetic  meridian.  Thus  in  1915  the  line  of 
zero  declination  passes  near  Columbns,  Ohio,  and  Charleston, 


FIG.  21. 


LAND  SUBVEYING. 


Oa.,  and  during  that  year  the  magnetic  meridian  coincided 
with  the  true  meridian  at  all  places  on  that  line.  Th.es:-: 
isogonic  lines  are  now  slowly  shifting  westward. 

The  secular  variation  of  the  magnetic  needle  is  an  oscillatory 
movement  by  which  the  declination  varies  back  and  forth  from 
a  mean  value.  The  time  of  this  oscillation  in  the  United  States 
is  between  two  and  three  centuries,  but  a  complete  cycle  has 
not  yet  been- observed.  For  example,  at  New  York,  N.  Y.,  the 
early  observations  indicate  that  in  1657  the  needle  was  at  its 
extreme  western  declination  of  9£  degrees  ;  this  slowly  de- 
creased so  that  about  1795  it  reached  the  minimum  value  of  4* 


1600 


1650 


1700 


ty 


1750     1800 
FIG.  22. 


1850 


1900 


1950 


Jegrees ;  during  the  nineteenth  century  it  has  slowly  increased 
and  will  probably  reach  the  extreme  western  declination  about 
1933,  the  total  period  of  the  cycle  thus  being  276  years.  Fig.  22 
shows  clearly  to  the  eye  these  variations  in  declination,  as  also 
those  at  Washington,  D.  C. ,  where  the  minimum  value  was  ob 
served  in  1810,  while  the  maximum  will  probably  occur  in  1927, 

The  value  of  the  declination  for  1915  may  be  ascertained 
approximately  from  the  isogonic  map  above  referred  to.  Its 
value  at  any  date  may  be  found  for  a  large  number  of  places 
by  means  of  the  formula?  deduced  by  the  U.  S.  Coast  and 
Geodetic  Survey,  and  given  in  the  report  for  1895,  pages  167 
to  320.  For  example,  the  formula  for  Bethlehem,  Pa.,  is 

D  =  5°.27  +  3°.05  sin  (l°.46m  -  34°.8), 


MAGNETIC   NEEDLE.  43 

in  which  D  denotes  west  declination  and  m  is  the  number  of 
years  counted  from  Jan.  1,  1850.  If  it  be  required  to  find  the 
declination  for  April  30,  1887,  the  value  of  m  is  37.3  years,  and 

then, 

D  =  5°.27  +  3°. 05  sin  19°.7  =  6°.50  west. 

From  the  formula  also  can  be  found  the  values  and  the  dates 
of  the  maximum  and  minimum  declinations.  The  greatest 
declination  will  occur  when  the  angle  l°.46w  —  34°. 8  equals  90°, 
as  the  sine  is  then  unity ;  this  gives  D  =  8°. 32  and  m  =  85.5 
years,  so  that  the  tiine  of  this  occurrence  will  probably  be  in 
the  year  1935.  The  least  declination  obtains  when  the  sine  is 
minus  unity,  and  this  gives  D  ~  2°. 22,  and  in  —  —  37.8,  which 
corresponds  to  the  year  1812. 

The  daily  variation  of  the  needles  is  a  small  oscillation  rang- 
ing from  5  to  10  minutes  in  different  seasons  and  places.  It  is 
smaller  in  the  winter  than  in  the  summer,  and  less  in  the 
southern  part  of  the  United  States  than  in  the  northern  part. 
Soon  after  sunrise  the  north  end  of  the  needle  is  at  its  most 
easterly  deviation  from  the  magnetic  meridian.  A  westerly 
motion  then  begins,  and  about  half-past  ten  o'clock  it  coincides 
with  that  meridian  ;  the  westerly  motion  continues  until  about 
half -past  one  o'clock  in  the  afternoon  when  the  most  westerly 
deviation  is  reached.  The  easterly  motion  is  then  slowly 
resumed  and  by  the  next  morning  the  needle  again  reaches  its 
most  easterly  deviation.  Table  III,  at  the  end  of  this  book, 
gives  the  mean  values  of  the  daily  variation  for  each  hour  of 
the  day  and  each  month  of  the  year  at  Philadelphia,  Pa.,  as 
also  instructions  for  finding  it  for  other  places  in  the  United 
States. 

In  addition  to  the  secular  and  daily  variations  the  magnetic 
needle  is  also  subject  to  an  annual  variation  of  about  \\  min- 
uteSj  and  to  other  smaller  variations  caused  by  the  moon  and 
sun.  Magnetic  storms  cause  sudden  variations  of  considerable 
amount.  These  minor  variations,  however,  are  of  little  im- 
portance in  land  surveying,  compared  to  the  local  attraction 
that  is  liable  to  occur  in  rocky  regions  and  which  often  causes 
discrepancies  of  several  degrees  in  the  bearings  of  a  line  taken 
at  points  only  a  few  hundred  feet  apart.  The  method  of 


44  LAND   SURVEYING. 

eliminating  the  effect  of  local  attraction  is  explained  in  the 
next  article. 

Prob.  13.  The  formula  for  the  west  declination  at  New 
Brunswick,  N.  J.,  is 

D  =  5M1  +  2°. 94  sin  (l°.30ro  +  4°.2). 

Find  the  values  of  the  maximum  and  minimum  declinations 
with  the  dates  of  their  occurrence.  Find  also  the  probable 
value  of  the  declination  on  June  15,  1896. 

ART.  14.    FIELD  WORK. 

The  field  work  in  land  surveying  may  be  divided  into  two 
classes,  original  surveys,  and  resurveys.  The  first  class  in- 
cludes not  only  the  case  of  lands  opened  for  the  first  time  for 
settlement,  but  also  the  staking  out  and  division  of  lands,  and 
all  surveys  which  are  made  without  particular  reference  to  for- 
mer records.  Resurveys,  on  the  other  hand,  are  those  made  to 
trace  boundaries  that  have  been  lost,  and  they  require  the 
knowledge  of  the  former  work  which  are  either  stated  in  deeds 
on  maps,  or  in  the  records  of  towns  or  counties.  In  both 
cases  the  field  work  requires  the  measurement  of  such  linns 
and  angles  as  will  enable  a  complete  map  of  the  property  to 
be  made,  and  the  areas  of  the  several  portions  to  be  computed,, 

A  field  party  usually  consists  of  three  or  four  men,  the  sur- 
veyor who  reads  the  angles  or  bearings  and  takes  the  notes, 
two  chainmen,  and  perhaps  an  axman  who  sets  the  necessary 
stakes  and  poles  and  also  assists  with  the  tape.  The  poles 
which  are  used  for  ranging  out  the  lines  and  to  sight  upon  in 
measuring  angles  are  generally  about  an  inch  in  diameter, 
about  eight  feet  long,  each  alternate  foot  being  painted  red 
and  white,  and  they  are  pointed  with  steel  to  enable  them  to 
be  easily  set  in  the  ground.  In  surveying  a  field  it  is  an  old 
custom  for  the  party  to  go  around  the  boundaries  ' '  in  the  di- 
rection of  the  sun,"  that  is,  so  as  to  keep  the  field  on  the  right 
hand.  The  bearings  of  lines  can  thus  be  written  on  a  sketch 
in  a  natural  order  around  the  entire  circuit. 

It  frequently  happens  that  a  surveyor  is  obliged  to  employ  j4s 
chainmen  men  who  have  had  no  experience  in  such  work.  In 


FIELD   WORK. 


45 


this  event  it  is  well,  even  after  having  given  them  full  instruc- 
tions, that  he  should  be  constantly  with  them  for  several  hours 
in  order  to  ensure  that  the  proper  degree  of  precision  shall  be 
attained.  Chaining  indeed  is  far  more  difficult  to  do  accu- 
rately than  is  the  measurement  of  angles. 

The  point  where  a  transit  is  set  for  the  purpose  of  reading 
angles  is  called  a  station.  In  the  survey  of  a  field  the  corners 
are  also  often  called  stations,  these  being  the  initial  points 
from  which  the  linear  measurements  are  taken.  A  line  whose 
bearing  is  known  is  frequently  called  a  course. 

If  the  surveyor  is  provided  with  a  transit  it  is  advised  that 
angles  should  be  always  measured,  and  only  such  bearings  be 
taken  as  are  necessary  to  check  the  work  or  to  verify  former 
records.  If  he  has  only  a  compass  the  bearings  of  the  lines  must 
be  taken,  but  care  should  be  exercised  to  avoid  the  errors  due 
to  local  attraction.  Fortunately  the  influence  of  this  can  be 
eliminated  by  always  reading  the  back  bearings  of  lines  as 
well  as  their  forward  bearings.  In  doing  this  the  instrument 
should  be  set  at  the  ends  of  the  lines  so  that  the  back  bearing 
of  one  line  and  the  forward  bearing  of  the  next  one  may  be 
read  at  the  same  station.  The  bearings  at  one  point  being  as- 
sumed to  be  correct,  all  the  others  can  then  be  adjusted  so  as  to 
be  relatively  correct. 

As  an  example  of  the  elimination  of  the  effect  of  local  attrac- 
ion  let  the  bearing  of  AB  be  taken  at  A  in  Fig.  9,  and  also 
the  back  bearing  of  EA\  then  at  B  let  the  bearings  of  BA  and 
BC  be  taken,  and  so  on.  Let  the  results  obtained  be  those 
which  are  given  in  the  second  and  third  columns  of  the  table. 


Course. 

Bearing. 

Back  Bearing. 

Adjusted  Bearing. 

Azimuth. 

AB 
BO 
CD 
DE 
EA 

N  37°  15'  E 
S  78    08   E 
S  33    45  W 
N  14    37  W 
N82    30  W 

S  38°  00'  W 
N77    45  W 
N33    15  E 
S  15    30  E 
S  83    15  E 

N  37°  15'  E 
S  78    53  E 
S  32    37  W 
N  15    15  W 
N82    15  W 

37°  15' 
101    07 
212    3? 
344    45 

277    45 

Now  assume  that  there  is  no  local  attraction  at  A,  then  the 
bearing  of  AB  and  the  back  bearing  of  EA  are  correct.  To 
adjust  the  other  values  proceed  in  order  from  A  to  B;  at  B  the 


46  LAND   SURVEYING. 

result  38*  00'  is  45'  too  large,  lience  45'  must  be  subtracted 
from  all  SW  and  NE  lines  starting  from  B  and  the  same 
amount  must  be  added  to  all  SE  and  NW  lines;  thus  the  ad 
justed  bearing  of  BC  is  78°  53'.  Next  the  result  77°  45'  taken 
at  G  is  seen  to  be  1°  08'  too  small,  and  this  must  be  applied  to 
the  forward  bearing  of  CD,  giving  the  adjusted  bearing  as 
S  32°  37'  W.  Thus  proceeding,  the  adjusted  bearing  of  EA 
comes  out  N  82°  15'  W,  and  this,  being  the  reverse  of  the  back 
bearing  taken  at  A,  is  a  check  on  the  correctness  of  both  the 
field  work  and  the  adjustment. 

The  azimuth  of  each  line  is  easily  found  from  its  adjusted 
bearing.  If  the  meridian  be  taken  to  correspond  with  the 
magnetic  meridian  the  results  given  in  the  last  column  of  the 
table  are  the  azimuths.  They  are  found  by  adding  or  sub- 
tracting each  bearing  either  to  or  from  180°  or  360°,  as  the  case 
may  require. 

The  interior  angles  of  a  field  are  readily  computed  either 
from  the  adjusted  bearings  or  from  the  azimuths  of  the  lines. 
It  is,  however,  no  proof  of  the  correctness  of  the  field  work  if 
the  sum  of  these  angles  equals  the  proper  theoretic  sum,  for 
it  will  be  found  that  any  bearings  whether  correct  or  incor- 
rect will  give  the  correct  amount.  On  the  other  hand  if  the 
angles  be  measured  in  the  field  with  the  transit,  a  valuable 
check  is  obtained  by  taking  their  sum  which  will  only  equjil 
the  theoretic  sum  in  very  good  work.  In  such  cases  if  no 
serious  error  is  thought  to  exist  the  observed  values  should  be 
adjusted  by  the  method  of  Art.  10. 

One  of  the  most  important  details  of  the  field  work  is  the 
keeping  of  the  notes.  Nearly  every  surveyor  has  a  system  of 
his  own  for  recording  the  measurements  taken  in  the  field,  so 
no  one  method  can  be  said  to  be  the  standard  ;  the  essential 
point  is  that  they  shall  be  readily  legible  to  any  person  who  is 
to  use  them.  Better  results  will  probably  be  obtained  by  mak- 
ing a  sketch  in  the  field  book,  showing  objects  in  their  relative 
positions  and  having  the  dimensions  to  be  used  in  plotting 
marked  on  the  sketch  itself,  than  by  a  more  elaborate  system 
of  symbols  and  abbreviations. 

If  the  survey  covers  but  a  small  area,  as  one  or  two  lots  of 


SURVEY   OF   A   FARM.  47 

town  property,  all  the  notes  should  be  recorded  on  one  sketch, 
which  may,  to  make  the  scale  larger,  be  extended  across  two 
pages.  In  the  survey  of  a  large  tract  it  will  be  better  to  devote 
a  page  to  one  course;  repeating,  as  the  leaves  are  turned,  part 
of  the  notes  of  one  page  on  the  next. 

The  notes  should  be  made  with  a  medium  hard  pencil  and  a 
straight-edge  be  used  in  drawing  all  lines  intended  to  be  straight. 
All  writing  should  be  in  upright  capitals,  and  no  script  should 
be  used.  Distances  along  the  line  are  usually  inclosed  in  a 
circle  or  parenthesis,  and  are  written  on  a  line  perpendicular 
to  the  base.  It  will  be  generally  more  convenient  to  begin  the 
notes  at  the  foot  of  the  page,  as  by  so  doing  one  can  glance 
from  the  book  to  the  field  and  see  corresponding  lines  having 
the  same  direction  and  in  front.  Samples  of  field  notes  are 
given  in  Art.  15.  The  best  books  for  notes  have  both  sides  of 
the  leaves  ruled  alike  with  light-blue  lines  into  squares  about 
an  eighth  of  an  inch  on  a  side.  Such  books  are  substantially 
bound  in  leather  and  cost  about  fifty  cents. 

Prob.  14.  Find  the  adjusted  bearings  of  the  sides  of  the 
following  field,  assuming  the  bearing  of  BC  to  be  correct. 

Length 
in  Chains. 

5.62 
'      3.28 
2.24 
3.05 
2.29 
6.40 
Also  compute  the  area  of  the  field  in  acres,  roods,  and  rods. 


ART.  15.     SURVEY  OF  A  FARM. 

Fig.  31  is  a  reduced  copy  of  a  farm  map  plotted  from  the 
field  notes  of  a  survey.  The  farm  is  seen  to  comprise  three 
divisions  separated  from  each  other  by  fences,  and  it  is  desired 
to  locate  the  interior  division  lines  as  well  as  the  boundaries, 
and  also  to 'mark  the  edge  of  the  wood-land  and  the  course  of 
the  brook. 

The  principal  lines  of  the  survey,  usually  called  traverse- 


Course. 

Bearing. 

Back  Bearing. 

AB 

S   12°  15'  W 

N  12°  30'  E 

BG 

N  76  45  W 

S  76   45  E 

CD 

N  12   15  W 

S  12   07  E 

DE 

N  47   37  W 

S  48  00  E 

EF 

N  24  30  E 

S  24  15  W 

FA 

S  75   15  E 

N  75  00  W 

48  LAND   SURVEYING. 

lines,  are  measured  outside  or  inside  the  boundaries  according 
to  circumstances;  thus  it  is  natural  that  measurement  along 
the  highway  should  be  easier  than  along  the  inside  of  the  fence, 
while  another  line  might  be  more  easily  measured  inside  the 
boundary  when  the  ground  is  there  clear  from  trees.  These 
traverse-lines  should  always  be  parallel  and  near  to  the  boun- 
dary lines  so  that  the  lengths  of  the  latter  may  be  obtained 
with  precision. 

The  manner  of  keeping  the  field-notes  is  shown  in  the  fol- 
lowing sketches  (Figs.  23-30).  On  the  first  page  of  the  note- 
book is  given  the  date  of  the  survey,  the  names  of  the  surveyor 
and  all  his  assistants,  and  also  a  sketch  of  the  traverse-lines 
with  letters  at  each  station  for  the  purpose  of  reference.  On 
the  second  and  succeeding  pages  of  the  note-book  are  the  notes 
of  the  traverses.  These  are  made  by  beginning  at  the  bottom 
of  the  page  and  working  upward,  so  that  the  surveyor  always 
has  the  objects  in  the  same  relative  position  as  the  sketches. 

The  survey  is  begun  by  setting  the  transit  over  B  and  select- 
ing stations  A  and  D.  The  interior  angle  ABD  is  read  and 
recorded  on  the  margin  of  the  page,  and  as  a  check  the  exterior 
angle  is  also  measured  and  written  under  the  first;  if  the  sum 
of  the  two  angles  is  within  one  minute  of  360  degrees,  the  first 
angle  is  recorded  on  an  arc  between  A  B  and  BD,  as  shown  in 
Fig.  24;  if  such  agreement  does  not  occur,  the  angles  should 
be  observed  again.  The  chain  is  then  drawn  from  A  to  Bt 
and  offsets  taken  with  the  tape  to  locate  the  ends  of  the  boun- 
dary line  and  the  corners  of  the  buildings;  the  sides  of  the 
buildings  and  the  width  of  the  highway  are  also  measured 
with  the  tape.  The  distances  from  A  along  the  traverse  are 
noted  opposite  to  each  offset,  and  the  offsets  themselves  are 
always  measured  perpendicular  to  the  traverse-line.  The 
magnetic  bearing  of  AB  is  taken  and  recorded  on  it,  while  the 
length  of  the  boundary  line  is  seen  from  the  distances  noted 
opposite  the  offsets  taken  at  its  ends. 

The  instrument  is  now  carried  forward  to  Dt  where  the  angle 
BDE  is  measured,  and  then  the  traverse-line  DE  is  run  par. 
%llel  to  the  next  side  of  the  field.  Thus  the  traverse-line* 


SURVEY   OF   A   FARM. 


Farm  of  George  Webster 
Riverside,  Pa. 

Surveyed  by 

John  Doe,  C.E. 

September  15,  1900. 

Jas.  Flynn 
Wm.  Roe 

A.Webster,  Axeman. 

Declination  of  Magnetic 
Needle  7°04' W. 


50 


LAND  SURVEYING. 


SURVEY   OF  A   FARM. 


51 


FIG. 


F».  28. 


LAND   SURVEYING. 


Fw.80. 


SUBVET  OF  A  FARM.  53 

around  the  farm  complete  the  polygon  ABDEFGHIKLAfA, 
and  the  interior  angles  of  this  polygon  should  equal  twice  as 
many  right  angles  as  the  polygon  has  sides  minus  four  right 
angles.  A  page  of  the  note-book  should  be  assigned  to  the 
description  of  some  of  the  principal  stations  or  corners  of  the 
farm,  so  that  they  may  be  found  in  case  of  a  resurvey.  The 
names  of  the  owners  of  the  adjoining  fields  should  also  be 
ascertained  and  recorded.  The  secondary  traverse  COVG  is 
run  to  locate  the  edge  of  the  woods,  while  OPQ  and  OUST 
locate  the  brook  and  the  pond. 

Great  care  should  be  taken  to  make  the  field-notes  clear  and 
complete  so  that  they  may  be  plotted  by  a  person  who  has  not 
seen  the  farm.  In  the  above  notes  five  angles  were  inadver- 
tently omitted  in  Fig.  29;  their  values  are  ITS=  114°  00*, 
TSR  =  220°  15',  SEO  =  144°  3(X,  HOP  =  230°  30',  and 
OPQ  =  220°  00'.  Magnetic  bearings  should  be  taken  on  at 
least  two  of  the  traverse-lines,  back  and  front  readings  being 
made  so  as  to  detect  any  local  attraction.  The  surveyor  should 
remember  that  the  notes  should  not  only  be  sufficient  to  plot 
and  describe  the  boundaries  of  the  farm,  but  also  be  so  com* 
plete  that  the  area  of  each  part  or  lot  can  be. computed. 

Prob.  15.  Find  the  bearings  and  lengths  of  each  of  the 
lines  of  the  closed  traverse  MNIKLM  from  the  field-notes 
in  Figs.  23-30,  and  compute  its  area. 

ART.  16.    OFFICE  WORK. 

Office  work  embraces  computations  and  the  drawing  of 
maps.  The  method  of  computing  the  area  of  a  polygon  has 
been  explained  in  Art.  6.  It  is,  however,  rarely  practicable  to 
have  the  lines  of  the  survey  coincide  with  the  boundaries  of 
the  field  or  farm,  and  hence  the  areas  of  the  trapezoids  between 
the  offsets  are  to  be  separately  computed  as  explained  in  Art. 
3,  and  these  are  added  to  or  subtracted  from  the  area  of  the 
polygon,  as  the  case  may  require.  All  computations  should  be 
checked  so  that  the  results  may  be  relied  upon. 

In  order  to  facilitate  the  work  of  plotting  the  map  the  lati- 
tudes and  longitudes  of  the  principal  stations  are  often  com- 


LAND   SURVEYING. 


8  Acres,  152  Rod 


FIELD 

7Acr.es,  127  Rods. 


MEADOW 
6  Acres,  1&  Rods. 


MAP  OF 

FAKM  OF  GEORGE  WEBSTEK, 
KIVEKSIDE,  PA. 

Surveyed  September  15^  1900, 

SCALE  OF  FEET  BY 

"o       100      200      300  JOHN  DOE,  0.  B. 

FIG.  31. 


OFFICE   WORK.  55 

puted.  For  example,  in  Art.  6,  Fig.  10,  it  is  most  convenient 
to  take  the  point  A  as  the  origin  of  coordinates.  The  latitude 
and  longitude  of  B  are  then  the  same  as  the  latitude  and  longi- 
tude differences  of  AB.  For  the  station  C  and  D, 

Lat.  G  =  799.94  +  249.98  =  1049.92 
Long.  G  =  0.00  +  433.07  =  433.07 
Lat.  D  =  1049.92  -  84.53  =  965.39 
Long.  D  =  433.07  +  181.29  =  614.36 

and  in  like  manner  the  latitude  and  longitude  of  each  station 
is  found  from  those  of  the  preceding  station  by  simply  adding 
or  subtracting  the  adjusted  latitude  and  longitude  differences 
of  the  line. 

To  plot  the  field  to  a  suitable  scale,  one  of  two  methods  is 
pursued  :  the  sides  of  the  polygon  are  laid  off  in  succession  by 
the  angle  with  the  preceding  course,  and  the  length  of  the 
course;  or  each  corner  is  located  independent  of  all  the  others 
by  means  of  its  previously  computed  co-ordinates. 

In  plotting  by  the  first  method  the  angles  are  laid  off  either 
by  the  protractor,  or  by  their  natural  sines  or  tangents.  Be- 
fore using  the  protractor  the  azimuths  of  all  the  courses  with 
reference  to  any  one  of  them  are  computed.  The  direction  of 
this  course  is  drawn  and  the  protractor  is  placed  in  position 
upon  it  and  fastened;  all  the  azimuths  are  pricked  off  around 
(he  edge  of  the  protractor  and  the  latter  is  removed.  The  di- 
rections of  all  the  courses  have  now  been  plotted  and  they  may 
be  transferred  to  any  part  of  the  paper  by  using  triangles. 
The  direction  of  any  course  as  AB  is  drawn  in  the  desired 
position  on  the  paper  and  its  length  measured  by  the  proper 
scale;  the  direction  of  BG  as  determined  by  the  protractor  is 
transferred  till  it  passes  through  B,  and  the  position  of  station 
(J  found  by  measuring  on  this  line  the  length  of  BG.  In  like 
manner  all  the  courses  are  plotted  and  the  accuracy  of  the  work 
is  proved  if  the  point  A,  plotted  in  order  after  the  others, 
coincides  with  the  position  assumed  for  it  at  first. 

To  lay  off  an  angle  by  means  of  its  natural  sine  an  arc  is 
drawn  whose  radius  is  10  on  any  scale.  A  chord  to  this  arc 
whose  length  is  the  sine  of  half  the  angle,  measured  with  a 


56  LAND  SURVEYING. 

scale  twice  as  large  as  before,  will  subtend  the  angle  at  the 
center.  Thus  to  plot  the  angle  ABC  of  40°,  with  B  as  a  center, 
an  arc  is  drawn  with  a  radius  of  10  to  the  scale  of,  say,  20  feet 
to  the  inch  ;  with  the  intersection  of  this  arc  and  AB  as  a 
center  strike  an  arc  with  a  radius  3.42  on  the  scale  of  10  feet 
to  the  inch,  cutting  the  first  arc  at  (7,  then  ABC  is  the 
required  angle. 

To  plot  the  same  angle  by  using  its  tangent,  mark  a  distance 
10  to  any  convenient  scale  from  B  toward  A\  at  that  point 
erect  a  perpendicular,  whose  length  is  8.39  to  the  same  scale, 
to  C,  and  ABG  is  the  angle  desired. 

The  first  method  of  plotting  a  map  has  the  merit  of  being 
easy  and  rapid,  but,  as  each  point  is  established  with  reference 
to  the  preceding  one,  any  error  in  the  location  of  a  station  will 
affect  the  position  of  all  that  are  fixed  after  that  one,  and  it  is 
to  overcome  this  difficulty  that  the  method  by  co-ordinates  if* 
used. 

After  the  coordinates  of  the  stations  have  been  computed  by 
taking  the  algebraic  sum  of  the  latitude  and  longitude  projec- 
tions of  the  preceding  courses,  the  origin  and  axes  of  coordi- 
nates are  plotted  upon  the  paper.  If  the  map  is  a  large  one  the 
utmost  care  must  be  taken  to  make  the  angle  between  the  axes 
exactly  90°  ;  the  right  angle  is  first  drawn  in  the  usual  way 
and  then  verified  by  measuring  the  hypothenuse  of  the  tri- 
angle as  large  as  the  limits  of  the  drawing  will  allow.  Paral- 
lel to  these  axes  lines  are  drawn  dividing  the  paper  into 
squares  100  feet,  200  feet,  or  1000  feet  on  a  side,  according  to 
the  scale  of  the  drawing,  the  object  being  to  bring  every  point 
on  the  map  within  the  length  of  the  scale  from  two  of  these 
lines.  The  stations  may  now  be  located  by  measuring  their 
coordinates  from  the  nearest  parallels  and  the  accuracy  tested 
by  the  length  of  the  sides.  In  plotting  the  houses,  fences,  and 
brooks,  the  scale  is  placed  on  the  traverse-line  and  all  the  dis- 
tances along  its  length,  to  points  where  offsets  are  taken, 
are  measured  without  moving  it ;  the  offsets  are  then  measured 
and  the  figures  completed. 

The  finished  map  should  contain  full  information  concerning 
the  date  of  survey,  scale  of  map,  names  of  owners  of  adjoining 


OFFICE   WOEK.  57 

property,  and  of  the  surveyor  ;  if  a  portion  of  the  plan  has 
been  compiled  from  other  maps  that  fact  should  be  stated  and 
references  given.  The  title,  meridian  point,  and  border  are, 
in  a  measure,  an  opportunity  for  the  exercise  of  artistic  skill 
on  the  part  of  the  draftsman,  but  legibility  and  simplicity 
must  not  be  sacrificed  for  ornament.  A  title  of  Roman  letters, 
well  done,  always  presents  a  good  appearance,  and  without 
other  decoration,  will  be  in  good  taste  on  maps  both  large  and 
small.  The  meridian  is  usually  represented  by  an  arrow  hav- 
ing the  head  at  the  north  end,  and  by  an  elongated  S  at  the 
south  ;  the  lines  should  be  very  light,  that  the  direction  may 
be  well  defined.  When  both  the  true  and  magnetic  meridians 
are  shown,  the  former  is  represented  by  a  full  arrow  and  the 
latter  by  one  having  but  one  side  of  the  head  drawn.  The  ap- 
pearance of  the  border  is  sometimes  improved  by  geometrical 
figures  or  some  simple  ornament  in  the  corners,  but  a  departure 
from  the  practice  of  using  simply  a  light  line  on  the  inside  and 
a  heavy  one  outside,  with  a  space  between  them  as  wide  as  the 
heavy  line,  will  be  for  the  worse  oftener  than  for  the  better. 

Prob.  16.  Compute  the  coordinates  of  the  stations  for  Fig. 
\33,  and  plot  the  map  of  the  farm  on  a  scale  of  100  feet  to  one 
Inch. 

ART.  17.    RANDOM  LINES. 

A  random  line  is  a  line  run  out  in  order  to  find  a  lost  corner, 
or  to  locate  a  boundary  line  which  has  become  obliterated. 
For  example  in  Fig.  32,  let  A  be  a  given 
corner  and  let  it  be  known  from  an  old 
record  that  a  certain  line  AP  was  once 
established  having  a  bearing  N  41°  30' 
W  and  a  length  of  32  chains.  No  traces 
of  this  line  or  of  the  corner  P  are  now 
visible,  and  it  is  required,  if  possible,  to 
relocate  them.  Between  the  date  of  the 
old  survey  and  the  present  one  the  decli- 
nation of  the  needle  has  changed  several  Fia.  32. 
degrees,  perhaps,  and  the  first  duty  of  the  surveyor  is  to  con- 
sider this  question  carefully  and  ascertain  the  probable  amount 


58  LAND   SURVEYING. 

of  change,  so  as  to  determine  the  present  probable  bearing  of 
the  line.  Suppose  that  the  result  of  this  inquiry  leads  to 
N  38°  15'  W  as  this  bearing. 

Starting  at  the  marked  corner  A  the  surveyor  runs  a  random 
line  AB  on  the  bearing  N  38°  15'  W,  and  measures  along  that 
line  a  distance  of  32  chains,  or  2112  feet,  to  a  point  B.  He 
then  proceeds  to  look  over  the  ground  on  both  sides  of  B  for 
the  lost  corner,  which  is  described  in  the  old  record  as  a 
marked  tree,  a  stump,  a  pile  of  stones,  or  a  monument.  If  it 
is  impossible  to  find  a  trace  of  it  nothing  further  can  be  done 
from  the  data  in  hand.  If,  however,  it  is  found  at  P,  a  per- 
pendicular PE  is  dropped  upon  the  line  AB  and  its  length  is 
measured,  as  also  the  distance  BE.  The  distance  AE  is  thus 
known,  and  from  the  right  triangle  the  angle  EAP  can  be  com- 
puted and  the  present  magnetic  bearing  of  AP  be  determined. 
For  example  :  Suppose  that  PE  is  found  to  be  37.4  feet, 
while  AEis  2110.5  feet,  then 


tan  EAP  =         ^-j  0.01772, 

whence  EA  P  =  1°  01',  and  hence  the  present  magnetic  bearing 
of  AP  is  N.  39°  16'  W.     The  distance  AP  is 


AP=  =  2110.8  feet, 

cos  1  01 

which  indicates,  if  the  present  work  is  accurate,  that  the  old 
survey  was  in  error  by  1.2  feet,  However,  it  is  a  principle  of 
law  that  established  corners  and  monuments  must  control  re- 
surveys,  and  hence  the  new  record  for  the  line  AP  is  N  39* 
16'  W  2110.8  feet. 

Intermediate  points  on  the  line  AP  may  now  be  established 
by  starting  at  A  and  running  it  out  with  the  new  bearing,,  A 
quicker  way,  however,  is  to  lay  off  perpendiculars  from  the 
stakes  previously  set  on  the  line  AE,  marking  their  lengths 
proportional  to  the  distances  from  A.  For  instance,  it  it  be 
required  to  mark  a  point  at  the  middle  of  AP,  the  perpendicu- 
lar to  be  erected  at  the  middle  of  AE  will  be  18.7  feet  in  length. 

Random  lines  are  also  frequently  used  to  find  the  bearing 
and  distance  between  two  points  which  are  not  intervisible, 


RESURVEYS.  59 

For  example,  let  G  and  H  in  tig.  33  be  two  such  points. 
Starting  at  G  let  a  line  GA  be  run  in  a  direction  which  is  ap- 
proximately toward  H.     On  arriving  at 
Ay  where  H  can  be  seen  let  AH  be  run. 
(Suppose  that  GA  is  N  42°  15'  E,  714.5 
feet;  and  that  AH  is  N  1°  08'  W,  210.5 
feet.     It  is  required  to  find  the  length 
and  bearing  of  GH. 

For  this  purpose  the  length  of  each 
line  is  multiplied  by  the  sine  and  cosine 
of  its  bearing,  and  the  results  tabulated 
as  below.  The  principle  that  the  sum 
of  the  northings  equals  the  sum  of  the  southings,  and  the  sum 
of  the  eastings  equals  the  sum  of  the  westings  (Art.  7),  gives 
739.4  feet  for  the  southing  of  HG  and  476.2  feet  as  its  westing. 
Dividing  the  second  of  these  by  the  first  gives  the  tangent  of 
Course.  Bearing.  Length.  Northing.  Southing.  Easting.  Westing. 
GA  N42C15'E  714.5  528.9  480.4 

AH    N    1   08  W     210.5      210.5  4.2 

HG  (739.4)  (476.2) 


739.4       739.4       480.4       480.4 

the  angle  between  HG  and  the  meridian,  while  the  square 
root  of  the  sum  of  their  squares  is  the  length  of  HG.  Thus 
the  bearing  of  HG  is  S  32°  47'  W,  and  that  of  GIlis  N  32°  47' 
E,  while  the  length  is  879.5.  This  length  can  also  be  found 
by  dividing  739.4  by  the  cosine  of  32°  47',  or  by  dividing  496. 2 
by  the  sine  of  32°  47'. 

Prob.  17.  In  order  to  find  the  direction  and  distance  be- 
tween two  points  TiTand  L,  the  following  lines  are  run  :  KA, 
S  87°  37'  W,  930.57  feet ;  AB,  West,  621.03  feet  ;  BL, 
S  88°  15'  W,  82.78  feet.  Compute  the  bearing  and  length  of 
KLt  and  locate  the  point  where  it  crosses  AB. 

ART.  18.    RESTJRVEYS. 

When  several  lines  of  the  boundary  of  a  farm  or  town  have 
become  obliterated  and  the  corners  lost,  it  is  often  necessary 
to  make  a  resurvey  in  order  to  re-establish  them.  If  the  corners 


60  LAND   SURVEYING. 

can  be  found  or  be  located  by  reliable  evidence  they  must  b* 
accepted  as  correct  even  if  the  recorded  bearings  and  lengths 
of  the  lines  indicate  different  points.  It  sometimes  happens 
that  some  corners  can  be  found  while  others  cannot.  In  such 
cases  a  series  of  random  lines  is  to  be  run  with  the  old  bear- 
ings, or  with  the  old  bearings  corrected  for  the  change  in 
declination  of  the  needle  between  the  two  dates. 


N 


FIG.  34. 

As  an  example  let  the  records  in  an  old  deed  give  the  bear 
ings  and  lengths  of  three  lines  as  follows: 

Ab,  N  60°  E,  10  chains; 

be,  N  45  E,  4  chains; 

cd,  S  45  E,  8  chains. 

There  being  no  definite  data  at  hand  to  determine  the  change 
in  magnetic  declination  between  the  dates  of  the  two  surveys, 
the  lines  AB,  BC,  and  CD,  are  run  with  the  given  bearings 
and  distances  from  the  known  corner  A.  The  old  corners  6 
and  c  cannot  be  found,  but  on  arriving  at  D  the  old  corner  d  is 
discovered  at  a  point  distant  20.4  links  and  S  12°  W  from  D. 
It  is  required  to  locate  the  old  corners  b  and  c. 

By  the  method  explained  in  Arts.  7  and  17,  the  bearings  and 
the  lengths  of  the  lines  DA  and  dA  may  be  computed.     These 

are : 

DA,  S  82°  47'  W,  17.29  chains; 

dA,  S83  26  W,  17.22  chains. 

Now  the  error  Dd  between  the  two  corners  is  due  to  two 
causes  •.  first,  to  a  constant  difference  in  the  magnetic  bear- 


BESURVEYS.  61 

ings  of  the  two  surveys;  and  second,  to  a  difference  in  the 
lengths  of  the  chains  used.  The  first  cause  swings  the  polygon 
AbcdA  around  the  point  A  by  a  small  angle.  The  second 
cause  alters  the  lengths  of  the  sides  in  a  constant  ratio.  The 
difference  between  the  bearings  of  DA  and  dA  is  the  constant 
angle,  while  the  ratio  of  the  lengths  of  these  lines  is  the  con- 
stant ratio.  To  find  the  bearings  of  the  old  lines,  therefore, 
each  of  the  given  bearings  is  to  be  corrected  by  the  amount 

83°  26'  ~  82°  47'  =  0°  39', 

and  to  find  the  lengths  of  the  old  lines  each  of  the  given 
lengths  is  to  be  multiplied  by 

•  S3-"-- 

All  of  this  reasoning  supposes  that  the  new  work  is  done 
with  such  precision  that  the  errors  in  chaining  must  be  re- 
g  arded  as  being  in  the  old  survey. 

Applying  these  corrections  the  adjusted  bearings  and  lengths 
of  the  old  lines  are 

Ab,  N    60°  39'    E,  9.96  chains; 

6c,  N    45  39      E,  3.99  chains; 

ed,  S    44  21      E,  7.97  chains, 

and  with  these  new  data  the  lines  may  be  rerun  and  the  cor- 
ners b  and  c  be  located,  a  check  on  the  field  work  being  that 
the  last  line  should  end  exactly  at  the  old  corner  d. 

It  is,  however,  not  difficult  to  compute  the  lengths  and 
bearings  of  Bb  and  Cct  so  that  b  and  c  may  be  located  from  the 
points  B  and  G.  The  principle  for  doing  this  is  that  the  poly- 
gons ABCDA  and  AbcdA  are  similar.  Thus  the  triangles 
ABb  and  ADd  are  similar;  hence  the  length  of  Bb  is 

D,       n  ,  AB      20.4  X  10 
1»=IW_=___ 

Also  the  angle  ABb  equals  the  angle  ADd,  or  70°  47' ;  hence 
the  bearing  of  Bb  is  S  10°  47'  E.  In  like  manner,  the  triangle 
ACc  being  similar  to  ADd,  the  length  and  bearing  of  Co 
can  be  found,  the  length  and  bearing  of  AC  being  first  com- 
puted. The  distance  Co  is  16.4  links,  and  its  bearing  is 


62  LAND   SURVEYING. 

S  15°  03'  E.  f/he  lines  Bl  and  Cc  are  now  run  from  B  and  (/, 
and  thus  the  most  probable  location  of  the  old  corners  b  and  c 
is  made. 

Prob.  18.  The  record  of  an  old  survey  reads  as  follows 
Commencing  at  a  post  marked  No.  5  and  running  N  62°  E, 
14.00  chains,  to  a  stake  marked  A\  thence  running  N  43£  E, 
8.00  chains,  to  a  stake  B;  thence  N  5C  W,  12.00  chains,  to  a 
stake  C\  thence  N  72£°  E,  10.25  chains,  to  a  stake  D\  thence 
S  12°  W,  6.43  chains,  to  a  stone  marked  No.  3.  On  rerunning 
the  lines  the  end  of  the  last  one,  instead  of  being  at  the  stone 
No.  3,  was  0.62  chains  due  East  from  it.  Find  the  adjusted 
bearings  and  lengths  of  the  old  lines;  also  find  the  distance  and 
direction  from  each  station  of  the  new  survey  to  the  corre- 
sponding one  of  the  old  survey. 


ART.  19.    TRAVERSING. 

The  term  traverse,  which  was  originally  associated  with 
navigation,  is  in  common  use  by  surveyors  to  define  a  series 
of  lines  whose  lengths  and  relative  directions  are  known.  For 
example  in  Fig.  23  the  lines  TS,  8JR,  RPy  constitute  a  trav- 
erse run  for  the  purpose  of  locating  a  brook.  Traversing  is 
particularly  applicable  to  the  survey  of  long  and  circuitous 
routes  through  territory  presenting  natural  obstructions  to 
long  sights.  It  is  almost  univerally  adopted  in  filling  in  the 
interior  of  maps  which  are  based  upon  a  system  of  triangula- 
tion.  As  examples  of  traversing  may  be  mentioned  the  survey 
of  highways  and  railroads,  river  banks,  shores  of  lakes,  and 
property  boundaries.  In  the  United  States  Government  sur- 
veys, when  the  traverse  is  run  to  mark  the  division  between 
private  estates  and  a  body  of  water  retained  as  public  property 
it  is  called  a  Meander  Line. 

The  most  approved  method  of  running  a  traverse  is  that  in 
which  the  graduated  plate,  or  limb,  of  the  transit  is  so  set  at 
each  station  that  the  azimuth  of  each  line  there  observed  can 
be  directly  read.  If  the  survey  is  made  in  a  locality  where  no 
system  of  latitudes  and  longitudes  has  been  established,  the 
magnetic  meridian  may  be  taken  as  the  meridian  of  the  azi- 
muths. At  the  first  station  the  vernier  is  set  at  zero  and  by 


TRAVERSING. 


means  of  the  lower  motion  the  instrument  is  turned  so  that  the 
north  end  of  the  needle  points  to  the  N  on  the  compass  limb. 
The  lower  plate  being  then  clamped  the  upper  one  is  un- 
claniped;  now  if  a  sight  be  taken  at  any  object  the  reading  on 
tlie  vernier  will  be  the  azimuth  corresponding  to  the  bearing 
of  that  object.  The  last  sight  and  reading  taken  at  the  first 
station  is  toward  the  second  station  of  the  traverse  line.  The 
instrument  is  then  placed  over  the  second  station  and  the  ver- 
nier set  at  the  back  azimuth  of  the  first  station ;  the  azimuth 
of  any  line  from  the  second  station  will  now  correspond  with 
its  bearing  as  before.  The  readings  of  the  needle  are  recorded 
as  a  rough  check  on  the  azimuths,  with  which  they  should 
agree  to  the  nearest  eighth  of  a  degree. 

For  example,  at  the  station  A  let  the  bearing  of  AB  be 
£i  74°  15'  E,  and  let  its  azimuth  be  74°  15'.  On  placing  the 
ii  istrunient  at  B,  the  vernier  is  set  at  254°  15',  a  sight  taken  on 
A,  and  the  lower  plate  clamped.  The  azimuth  of  BG  being 
143°  02',  the  vernier  is  set  at  323°  02'  on  arriving  at  C  and  the 
li  mb  placed  in  proper  position  by  sighting  back  to  B-  The 
uJescope  is  not  reversed  during  any  part  of  the  work.  At 
eich  of  the  stations  sights  may  be  taken  to  surrounding  ob- 
jects, and  if  the  distance  to  an  object  is  measured  this  together 
vith  its  azimuth  locates  it  with  respect  to  the  station. 


Bearing. 

Azimuth. 

Distance. 

Object  Sighted. 

NOTES 

AT  STATION 

B 

S  74°  15'  W 
S  37°  00'  E 

254°  15' 
325    42 
196    24 
194    10 
143    02 

528.3 
250. 

490.7 

Station  A 
Large  pine  tree 
NE  corner  of  John  Doe's  House 
SE  corner  of  J.  Doe's  same  House 
Station  C 

NOTES 

AT  STATION 

C 

N  37°  05'  W 
S  42°  45'  E 

323°  02' 
280    13 
276    15 
104    07 
137    15 

490.7 

985 
504.6 

Station  B 
NE  corner  of  John  Doe's  House 
SE  corner  of  J.  Doe's  same  House 
Fence  corner 
Station  D 

The  field  notes,  if  offsets  are  taken  from  the  traverse  lines 
best  kept  as  in  Figs.  24-31,  the  bearing  of  a  line  being 
written  upon  one  side  of  it  and  the  azimuth  upon  the  other  side. 


64  LA1STD   SUKVEYIXG. 

If  no  offsets  are  taken  a  form  like  that  given  abov*  may  be 
used.  It  is  seen  that  the  large  pine  tree  is  located  by  azimuth 
and  distance,  at  station  B,  as  also  is  the  fence  corner  at  station 
C.  The  house  of  John  Doe,  however,  is  located  by  azimuths 
taken  from  both  B  and  C,  the  line  BC  forming  a  base  by  which 
its  distance  from  either  end  can  be  computed. 

It  is  always  desirable  that  a  traverse  should  have  a  check 
upon  its  accuracy.  In  a  closed  traverse  like  that  around  thf* 
boundaries  of  a  farm  this  is  obtained,  since  the  sum  of  thf 
northings  must  equal  the  sum  of  the  southings,  and  the  sum 
of  the  eastings  that  of  the  westings.  In  Fig.  23,  the  traverse 
CNOPQG,  which  begins  at  C  and  ends  at  6r,  is  checked  in  the 
field  on  arriving  at  6r,  for  the  azimuth  of  G^fiTrnust  agree  with 
that  previously  obtained ;  also  in  computation  the  differences 
of  latitude  and  longitude  between  C  and  G  must  agree  with 
those  obtained  from  the  main  polygon. 

It  should  be  remarked  that  the  object  of  taking  the  bearings 
is  merely  to  check  gross  errors  in  the  azimuths  during  the 
progress  of  the  field  work,  and  that  an  experienced  engineer 
will  usually  prefer  to  take  but  few  readings  of  the  needle.  If 
a  true  meridian  has  been  established  in  the  neighborhood  of 
the.  survey  the  azimuths  should  be  reckoned  from  it  instead  of 
from  the  magnetic  meridian. 

Prob.  19.  Compute  from  the  above  notes  the  length  of  the 
west  side  of  John  Doe's  house.  Obtain  the  same  distance 
Without  computation  by  plotting  the  notes. 

ART.  20.     UNITED  STATES  PUBLIC  LAND  SURVEYS. 

The  system  adopted  by  the  United  States  Government  on 
^ay  20,  1785,  for  the  survey  of  the  public  land  which  had 
been  acquired  from  time  to  time,  consists  in  dividing  it  into 
squares,  called  townships,  six  miles  on  a  side,  by  meridians 
and  east  and  west  lines.  A  north  and  south  row  of  townships 
is  called  a  range.  The  townships  are  divided  into  square 
miles,  called  sections,  which  are  subdivided  into  half  and 
quarter  sections. 

The  work  of  surveying  the  government  land  is  begun  by 


UNITED    STATES    PUBLIC   LAND   SURVEYS.         65 

carefully  running  a  north  and  south  line,  called  the  principal 
meridian,  and  an  east  and  west  line  called  the  standard  parallel. 
Standard  parallels  and  accurate  guide  meridians  are  run  to 
divide  the  territory  into  24  mile  squares,  and  the  principal 
meridians  are  at  long  intervals — 100  miles  or  more.  On  these 
lines  every  mile  is  marked  by  a  stake  or  monument  and  called 
a  section  corner;  every  sixth  section  corner  is  called  a  town- 
ship corner  and  is  differently  marked. 

On  the  standard  parallel  the  township  corners  are  next 
marked;  from  each  of  these  marks  range  lines  are  run  to  inter- 
sect the  standard  parallel  next  north.  Owing  to  the  con- 
vergence of  meridians  toward  the  pole,  the  points  of  their 
intersections  with  the  standard  parallel  will  not  be  at  the 
township  corners,  but  a  little  nearer  the  principal  meridian; 
as  the  full  six  miles  have  been  measured  on  the  standard  par- 
allels, the  convergence  is  corrected  at  each  of  those  lines. 

From  the  township  corners  on  the  principal  meridian,  east 
and  west  lines  are  run  joining  the  range  lines  already  fixed. 
The  townships  thus  marked  are  six  miles  north  and  south  by 
six  miles,  less  the  meridional  convergence  in  the  distance  to 
the  standard  parallel,  east  and  west. 

Parallel  to  the  eastern  boundaries  of  the  several  townships, 
section  lines  through  the  section  corners  are  run  for  five  miles, 
then  from  the  points  where  they  intersect  the  fifth  east  and 
west  section  lines,  oblique  lines  are  run  to  the  points  previously 
established  on  the  northern  boundary  of  the  -township;  when, 
however,  the  northern  boundary  of  the  township  is  one  of  the 
standard  parallels,  the  section  meridians  are  run  directly  the 
full  six  miles  instead  of  deflecting  at  the  fifth  east  and  west 
line. 

The  convergence  of  the  meridians  is  given,  very  nearly,  by 
the  following  rules  of  geodesy: 

The  angular  meridional  convergence  equals  the  difference 
in  longitude  into  the  sine  of  the  latitude. 

The  linear  convergence  equals  the  distance  along  the  meridian 
into  the  sine  of  the  angular  meridional  convergence. 

The  townships  are  divided  into  36  sections,  numbered  from 


66 


LAND  SURVEYING. 


1  to  36,  as  shown  in  Fig.  35.     The  sections  themselves  are 

subdivided  and  designated  as  in  Fig.  36  ;  a  represents  the  va- 
rious ways  of  dividing  an  entire 
section,  and  b  shows  the  method 
when  a  portion  of  the  section  is 
obstructed  by  water.  In  cases 
of  this  kind  it  is  usual  to  add  to 
an  adjacent  lot  the  salable  part 
of  the  obstructed  quarter  section, 
and  to  state  the  total  number  of 
acres  in  both;  but  when  only  a 
FlG-  85-  small  portion  of  the  quarter 

section  is  unsalable  it  retains  its  own  name,  is  called  fractional, 

and  the  number  of  acres  in  it  are  given. 


6 

5 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

80 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

North  # 

N.W.  M 
of 
S.W.  M 

E.  ^ 
of 
S.W.  H 

S.  E.  H 

S.W.^4 

or 

S.W.  K 

a  b 

FIG.  36. 

The  methods  of  running  the  principal  meridians  and  stand- 
ard parallels  are  founded  on  the  science  of  geodesy.  The  rules 
governing  the  running  of  township  and  section  lines  may  be 
found  in  "Instructions  to  the  Surveyors  General  of  Public 
Lands, "  issued  by  the  Land  Office  of  the  Interior  Department, 
Washington,  D.  C.  The  principles  of  this  chapter  and  the  last 
are,  however,  directly  applicable  to  the  surveying  and  map- 
ping of  townships,  sections,  and  their  subdivisions. 

Prob.  20.  Compute  the  length  of  the  northern  and  southern 
boundaries  of  a  township  in  latitude  46°  30',  the  southern 
boundary  being  18  miles  north  of  a  standard  parallel. 


THE  LEVEL.  67 


CHAPTEE  III. 

LEVELING  AND  TRIANGULATION. 
ART.  21.    THE  LEVEL. 

The  Engineer's  Level  consists  of  a  line  of  sight  parallel  to  a 
spirit  level  and  perpendicular  to  a  vertical  axis.  The  line  of 
siglit  is  fixed  in  a  telescope  by  cross-hairs  as  in]  the  transit. 
The  spirit  level  is  attached  to  the  under  side  of  the  telescope 
and  is  protected  except  on  top  by  a  rnetal  tube.  The  telescope 
is  supported  on  vertical  forks,  called  Ys  (from  which  fact  the 
instrument  is  called  the  Y  level),  and  is. clamped  to  them  by 
collars  which  may  be  raised,  allowing  the  telescope  to  be 
turned  on  its  axis  or  taken  out  entirely.  The  Ys,  which  may 
be  lengthened  or  shortened  by  screws  for  the  purpose,  are 
fastened  to  a  horizontal  bar  which  is  rigidly  attached  to  the 
vertical  axis.  The  instrument  is  provided  with  leveling 
screws  and  mounted  upon  a  tripod. 

The  Dumpy  Level  differs  from  the  ordinary  form  in  having 
the  telescope  firmly  fixed  on  the  horizontal  bar  so  it  cannot  be 
turned  either  on  its  axis  or  end  for  end.  This  level  is  superior 
to  the  Y  type  in  every  point  of  difference,  being  less  costly, 
lighter,  and  more  permanent  in  its  adjustment.  The  superior- 
ity claimed  for  the  Y  level  is  the  ease  of  adjustment  by  means 
of  its  movable  telescope,  but  if  such  an  advantage  exists  it  is 
extremely  slight. 

The  parts  of  the  level  of  most  importance  are  the  telescope 
and  the  bubble.  The  character  of  the  work  to  be  done  will 
determine  whether  or  not  magnifying  power  in  the  telescope  is 
more  desirable  than  illumination  of  the  field  of  view  and  what 
was  said  on  this  subject  in  connection  with  the  transit  applies 
as  well  to  the  level.  The  upper  part  of  the  inside  surface  of 
the  bubble  tube  is  carefully  ground  in  the  form  of  a  longitu- 
dinal circular  curve,  and  upon  the  radius  of  this  curve  depends 
what  is  known  as  the  sensitiveness  of  the  level.  If  the  radius 
of  curvature  of  the  bubble  is  large  it  will  be  very  sensitive ; 


68  LEVELING  AND  TRIANGULATION. 

that  is,  a  slight  vertical  displacement  of  the  telescope  will 
cause  a  considerable  motion  of  the  bubble.  If  the  radius  of 
curvature  is  short  the  bubble  is  not  sensitive.  A  very  sensi- 
tive bubble  is  not  desirable  since  much  time  will  then  be  lost 
in  leveling  the  instrument. 

The  level  rod  is  a  graduated  scale  for  measuring  the  vertical 
distance  between  the  horizontal  plane  through  the  line  of  sight 
and  that  through  the  point  upon  which  the  rod  is  held.  Tar- 
get rods  are  used  in  precise  work,  and  self-reading  rods  in 
cases  where  elevations  need  to  be  determined  only  to  tenths  of 
a  foot.  The  target  rod  has  a  vernier  on  its  movable  target  by 
which  readings  to  the  thousandth  of  a  foot  are  taken  by  the 
rodman  ;  the  New  York  rod,  the  Boston  rod,  and  the  Philadel- 
phia rod  are  the  most  common  forms  in  use.  Self-reading  rods 
have  figures  and  graduations  distinct  enough  to  be  read  by  the 
leveler  as  he  sights  through  the  telescope.  A  self-reading  rod 
is  divided  into  tenths  of  a  foot,  but  if  the  figures  are  properly 
made  readings  to  hundredths  of  a  foot  can  easily  be  taken  ;  the 
numbers  marking  the  tenths  should  be  0.06  feet  long  and  so 
placed  that  half  the  length  is  above  and  half  below  the  line. 
The  numbers  marking  the  feet  are  0.10  feet  long,  and  each  is 
bisected  by  the  foot-mark. 

Prob.  21.  Sketch  a  part  of  a  target  rod  showing  a  vernier 
reading  5.027  feet.  Sketch  a  self-reading  rod  according  to  the 
above  directions. 

ART.  22.    ADJUSTMENTS  OF  A  LEVEL. 

The  adjustment  of  an  instrument  consists  in  bringing  the 
various  parts  into  their  proper  relative  positions  so  that  all  the 
geometrical  conditions  necessary  for  good  work  may  be 
observed.  When  an  instrument  is  received  from  the  maker  it 
should  be  in  perfect  adjustment,  and  with  proper  c-are  it  will 
remain  so  for  a  long  time.  It  should,  however,  be  examined 
at  frequent  intervals,  and  if  found  out  of  adjustment  at  any 
time,  should  be  at  once  put  into  proper  condition.  The  fol- 
lowing description  of  the  adjustments  of  the  Y  level  follows  the 
order  in  which  they  should  be  made. 


ADJUSTMENTS   OF   A   LEVEL.  69 

Parallax. — 'This  is  an  improper  condition  of  focusing  due  to 
the  fact  that  the  image  does  not  fall  in  the  plane  of  the  cross- 
hairs. To  ascertain  if  it  exists,  direct  the  telescope  upon  the 
sky  and  focus  the  eyepiece  so  that  the  cross-hairs  are  perfectly 
distinct.  Then  turn  the  telescope  upon  the  object  which  is  tc 
be  observed,  and  focus  the  object  glass  until  the  image  is  per- 
fectly distinct.  Move  the  eye  from  side  to  side  and  note 
whether  there  is  any  apparent  movement  of  the  cross-hairs  and 
image.  If  any  is  seen  the  two  operations  are  to  be  repeated 
until  all  parallax  is  removed.  This  adjustment  depends  upon 
the  eye  of  the  observer,  and  when  made  for  one  person  may 
not  be  correct  for  another. 

Collimation. — The  line  of  signt,  or  collimation,  should  not 
deviate  from  the  optical  axis  of  the  telescope.  To  ascertain  if 
an  error  in  collimation  exists,  loosen  the  collars  on  the  Y's  and 
focus  the  telescope  upon  a  distant  object.  Slowly  revolve  the 
telescope  in  the  Y's  and  note  whether  the  intersection  of  the 
cross-hairs  remains  on  the  same  point.  If  the  horizontal  hair 
deviates  from  the  point  adjust  it  by  moving  it  over  half  the 
apparent  error,  by  means  of  the  capstan  screws  on  the  top  and 
bottom  of  the  telescope.  If  the  vertical  hair  deviates  adjust  it 
by  moving  it  over  half  the  apparent  error  by  means  of  the  cap- 
stan screws  on  the  sides  of  the  telescope.  The  instrument  is, 
of  course,  to  be  clamped  while  making  this  adjustment,  but  it 
need  not  be  leveled. 

The  Attached  Bubble.— The  level  bubble  attached  to  the 
telescope  must  be  parallel  to  the  line  of  sight.  To  ascertain  if 
this  is  the  case,  span  the  collars,  carefully  level  the  instrument 
and  clamp  it;  lift  the  telescope  out  of  the  Y's,  turn  it  end  for 
end,  and  replace  it.  If  the  bubble  does  not  settle  in  the 
middle  turn  the  screws  above  and  below  one  end  of  the  bubble- 
tube  -so  as  to  bring  the  bubble  half  way  back.  Next  see  if  the 
bubble  is  in  the  same  plane  as  the  telescope  by  slowly  revolv- 
ing the  latter  in  the  Y's  and  noting  whether  the  bubble  runs 
away  from  the  middle;  if  it  does  correct  half  the  apparent 
error  by  the  screws  on  the  sides  of  the  other  end  of  the  bubble- 
tube.  Repeat  these  operations  until  perfect  adjustment  is 
secured. 


70  LEVELING   AND   TKIANGULATION". 

The  Horizontal  Bar. — The  telescope  and  level-bubble  should 
be  parallel  to  the  horizontal  bar  supporting  the  Y's,  or  perpen- 
dicular to  the  vertical  axis  of  the  instrument.  To  ascertain  if 
this  is  the  case  after  the  preceding  adjustments  have  been 
made,  level  the  instrument  and  revolve  it  180  degrees  on  the 
vertical  axis.  If  the  bubble  runs  toward  one  end,  the  Y  on 
that  end  is  too  high,  and  the  screws  at  the  end  of  the  horizontal 
bar  are  moved  so  as  to  correct  one  half  of  the  apparent  error. 
Then  repeat  the  operation  until  the  bubble  remains  in  the 
middle  of  the  scale  for  all  positions  of  the  telescope. 

In  adjusting  an  instrument  great  care  must  be  taken  not  to 
turn  the  screws  too  tight,  as  by  so  doing  the  threads  soon  bs- 
come  injured.  No  student  or  beginner  should  be  allowed  to 
adjust  a  level  or  transit  until  he  has  become  well  acquainted 
with  all  its  parts  by  actual  use.  The  parallax  adjustment, 
however,  is  an  exception,  since  this  varies  for  different  eyes, 
and  each  student  should  see  that  this  is  made  every  time  he 
uses  the  instrument. 

The  dumpy  level  cannot  be  adjusted  by  the  above  methods 
since  the  horizontal  bar  and  telescope  are  rigidly  connected. 
Both  the  bubble  and  the  horizontal  cross-hair  are,  however, 
movable.  It  is  necessary,  (a)  that  the  bubble  should  be  per- 
pendicular to  the  vertical  axis  and  (b)  that  the  line  of  sight 
should  be  parallel  to  the  bubble.  The  adjustment  (a)  is  made 
exactly  like  that  above  described  for  the  horizontal  bar  of  the 
Y  level.  The  adjustment  (5)  is  made  by  the  peg  method  of 
Art.  26,  except  that  the  horizontal  cross-hair  is  moved  instead 
of  the  bubble. 

Prob.  22.  Give  the  reasons  for  each  of  the  adjustments  of 
the  Y  level. 

ART.  23.    COMPARISON  OF  LEVELS. 

In  buying  an  instrument  it  is  desirable  that  the  survey 01 
should  be  able  to  make  such  an  examination  as  will  indicate 
whether  it  is  a  good  one  of  its  class  or  whether  it  is  the  kind 
that  he  needs.  The  following  tests,  which  are  useful  in  addi- 
tion to  those  of  the  last  article,  will  be  found  valuable  in 


COMPARISON   OF   LEVELS,  71 

selecting  an  instrument,  or  in  comparing  one  with  another.  In 
making  them  the  instrument  should  be  in  good  adjustment. 

Magnifying  Power. — The  magnifying  power  of  a  telescope 
may  be  obtained  by  dividing  the  focal  length  of  the  object 
glass  by  that  of  the  eyepiece.  As  these,  however,  cannot  be 
closely  measured  the  following  method  is  usually  preferable: 
Place  a  rod,  on  which  the  divisions  are  very  plainly  marked, 
about  25  yards  from  the  instrument  and  focus  the  telescope 
upon  it.  Turn  the  line  of  sight  slightly  away  from  the  rod 
and  focus  the  other  eye  upon  it.  Slowly  turn  the  telescope 
again  toward  the  rod,  when  the  small  image  as  £een  by  that 
eye  will  appear  projected  upon  the  larger  one  seen  through 
the  telescope.  If,  for  instance,  100  divisions  seen  by  the 
naked  eye  appear  to  cover  5  divisions  seen  by  the  other  eye 
through  the  telescope,  then  the  magnifying  power  is  100  •*•  5 
=  20.  A  high  magnifying  power  implies  a  small  field  of  view 
and  hence  is  not  desirable.  For  a  surveyor's  transit  or  level  a 
magnifying  power  of  from  15  to  20  is  sufficient;  for  an  engi- 
neer's transit  it  should  be  from  20  to  25,  and  for  an  engineer's 
level  perhaps  from  25  to  30. 

Spherical  Aberration. — This  is  a  defect  caused  by  combin- 
ing lenses  of  different  curvatures,  so  that  objects  on  the  sides 
of  the  field  of  view  are  seen  less  distinctly  than  those  in  the 
center.  To  test  the  object  glass  for  this  defect,  cover  the  outer 
edge  with  an  annular  ring  of  paper  and  focus  upon  a  distant 
object;  then  remove  the  ring  and  cover  the  central  part  of  the 
glass;  if  no  change  of  focus  is  needed  the  glass  has  no  spheri- 
cal aberration.  To  test  the  eyepiece,  sight  to  a  heavy  black 
line  drawn  on  white  paper  and  held  near  the  side  of  the  field 
of  view;  if  it  appears  perfectly  straight  the  eye  glass  is  a  good 
one. 

Chromatic  Aberration. — This  is  a  defect  caused  by  com^ 
bining  lenses  of  improper  varieties  of  glass  so  that  yellow  01 
purple  colors  appear  on  the  edges  of  the  field.  To  test  a  tele- 
scope for  this  defect,  focus  it  upon  a  bright  distant  object  and 
slowly  move  the  object  glass  out  and  in;  if  no  colors  ar« 
observed  around  the  edges  of  the  field  of  view  the  telescope  is 
f re<»  from  this  defect. 


72  LEVELING   AND   TKIANGULATIGK. 

Definition. — Tlie  ability  to  show  images  with  sharp,  clean 
outlines  is  a  valuable  quality  in  a  telescope.  It  may  be  tested 
by  comparing  the  distinctness  of  the  image  with  that  of  the 
object  as  seen  by  the  eye  at  such  a  distance  that  it  will  seem 
the  same  in  size  as  the  image.  Ordinary  print  when  read  by 
the  eye  and  through  the  glass  with  equal  ease  should  appear 
equally  distinct. 

Size  of  Field. — The  angular  diameter  of  the  field  of  view  is 
usually  about  one  degree.  The  value  for  any  telescope  may  be 
closely  obtained  by  laying  off  a  distance  of  57.3  feet  from  the 
object  glass,  placing  two  pins  in  the  ground  at  the  extreme  sides 
of  the  field,  and  measuring  the  distance  between  them  in  feet; 
this  will  be  the  size  of  the  field  of  view  in  degrees.  (Art.  2.) 

Sensitiveness  of  Bubble. — For  very  fine  work  the  radius  of 
curvature  of  a  level  bubble  should  be  about  100  feet,  for  ordinary 
good  work  50  feet  is  preferable,  and  for  common  work  25  feet 
will  do.  To  determine  this  radius  let  the  instrument  be  set  up 
and  leveled,  so  that  two  screws  will  be  in  the  line  of  sight  to 
a  target  rod  placed  100  feet  or 
more  away.  Let  one  end  of  tbe 
bubble  be  made  to  coincide  with 
one.  of  the  division  marks  at  a  and 
a  reading  be  taken  on  the  rod  at  A. 
Then  by  the  two  screws  let  the  tel- 
escope be  raised  in  a  vertical  plane 
until  the  end  of  the  bubble  reaches 
FIG.  37.  the  next  division  at  5,  when  a 

second  reading  is  taken  on  the  rod  at  B.  Now,  if  R  be  the 
radius  of  the  level  bubble  and  D  the  distance  from  the  instru- 
ment to  the  rod,  R  :  D  : :  ab  :  AS  very  nearly.  The  distance 
AB  is  the  difference  of  the  readings  on  the  rod,  while  ab  is  the 
length  of  one  space  of  the  bubble  scale ;  thus  D  is  known.  For 
example,  let  the  rod  be  150  feet  from  the  instrument,  the  two 
rod  readings  be  3.704  and  3.745  feet,  and  the  bubble  scale  have 
8  spaces  in  one  inch,  one  space  thus  being  ^  of  a  foot  long. 
Then 


LEVELING.  73 

which  is  the  radius  of  the  level  bubble.  The  operation  should 
now  be  repeated  using  a  different  distance  D,  and  the  mean  of 
several  results  be  taken  as  a  final  value. 

Prob.  23.  A  level  bubble  has  a  radius  of  125  feet  and  its 
scale  has  10  spaces  in  an  inch.  What  error  in  leveling  will 
result  at  a  distance  of  250  feet  if  the  bubble  is  1£  spaces  out  of 
level  ? 

ART.  24.    LEVELING. 

A  Level  Surface  is  that  of  a  fluid  at  rest,  and  a  Level  Line  is 
the  intersection  of  such  a  surface  with  a  vertical  plane.  The 
line  of  sight  through  the  telescope  of  a  properly  leveled  and 
Adjusted  leveling  instrument,  when  revolved  around  the  verti- 
tal  axis,  generates  a  plane  which,  for  short  distances,  practi- 
cally coincides  \\4th  the  level  surface  through  the  instrument. 


FIG.  38. 

The  amount  of  deviation  between  the  two  surfaces,  due  to  the 
curvature  of  the  earth  and  to  refraction,  varies  as  the  square 
of  the  horizontal  distance  from  the  instrument  and  at  one  mile 
is  about  .57  feet. 

The  field  work  of  leveling  consists  in  finding  the  relative 
elevations  of  two  or  more  points.  The  elevations  are  referred 
to  an  assumed  surface  called  the  Datum  Plane,  or  simply 
Datum,  which  is'  so  selected  that  all  points  whose  elevations 
are  required  shall  be  above  it.  A  mean  sea  level  is  frequently 
taken  as  the  datum  plane.  A  Bench  Mark  is  a  monument, 
rock  or  other  permanent  object  whose  elevation  above  the 
datum  has  been  determined.  The  method  of  carrying  on  the 
field  work  can  best  be  explained  by  Fig.  38.  The  line  MN  rep- 
resents the  datum  plane ;  a  is  a  bench  mark  whose  elevation 
is  known ;  b,  c,  d,  e,  f,  are  points  whose  elevations  are  desired; 


74  LEVELING  AND  TRIANGULATION. 

A,  B,  and  C  are  the  successive  positions  of  the  instrument. 
The  positions  of  the  rod  are  indicated  by  the  vertical  lines  and 
the  lines  of  sight  by  the  horizontal  dotted  ones.  The  instru- 
ment is  leveled  at  A  and  the  reading  al,  on  the  bench  mark  at 
a,  is  taken;  this  is  called  a  Back  Sight  and  is  added  to  the  eleva- 
tion Ma,  to  get  the  Height  of  Instrument.  The  rod  readings  at 
6,  c,  and  d,  subtracted  from  the  height  of  instrument  will  give 
the  elevations  of  those  points  above  the  datum  MN ;  such  read- 
ings are  called  Fore  Sights.  If  the  distance  Ad  is  as  far  as  can 
be  seen,  the  rod  is  kept  at  d,  which  is  called  a  Turning  Point;  the 
instrument  is  carried  forward  to  B,  and  the  back  sight  dn  is 
taken ;  the  new  height  of  instrument  is  then  Pd  -\-  dn,  and  fore 
sights  at  e  and  /,  are  taken  to  determine  the  elevations  of  the 
stations  e  and  /.  The  instrument  may  then  be  carried  forward 
to  £7  and  the  elevations  of  g,  h,  and  k  determined  in  a  similar  man- 
ner. If  the  instrument  is  always  set  midway  between  the  turn- 
ing points,  the  errors  in  rod  readings,  due  to  the  non-adjust- 
ment of  the  instrument  and  to  the  curvature  of  the  earth,  will 
be  confined  to  the  intermediate  points  as  6,  c,  and  e\  this  fact 
should  always  be  remembered  as  upon  it  depends,  in  a  great 
measure,  the  accuracy  of  the  work.  The  turning  points  are 
not  necessarily  taken  at  places  whose  elevation  is  desired,  bnt 
may  be  at  any  convenient  location,  either  on  or  off  the  lines; 
they  should  be  so  selected  that  an  unobstructed  view  of  the  rod 
may  be  had  from  any  probable  position  which  may  be  selected 
as  the  next  place  for  the  instrument,  and  be  upon  firm  objects 
which  cannot  be  readily  disturbed  while  the  instrument  is  be- 
ing carried  forward. 

The  field  notes  are  kept  as  shown  below;  they  are  usually  on 
the  left-hand  page  of  the  note  book  while  the  opposite  page  is 
devoted  to  remarks.  The  first  column  gives  the  name  or  num- 
ber of  the  point  where  the  rod  is  placed;  such  a  point  is  called 
a  Station.  If  the  stations  are  in  a  continuous  line,  as  along  the 
middle  of  a  road,  the  distances  between  them  are  given  in  the 
second  column.  The  back  sights  are  given  in  the  next  column; 
then  the  height  of  instrument,  foresight,  and  elevation,  in  the 
order  named.  This  arrangement  will  be  found  most  conven- 
ient in  making  the  additions,  for  the  height  of  instrument  and 


CONTOURS   AND   PROFILES. 


75 


the  subtractions  for  the  elevations.  It  is  seen  £hat  the  rod  is 
read  to  thousandths  of  a  foot  on  the  bench  marks  and  turning 
points  and  to  hundredths  of  a  foot  on  the  other  points.  In 
work  of  less  precision  than  that  in  towns  and  cities  the  rod 


Station 

Dist. 

B.S. 

H.I. 

F.S. 

Eleva. 

Remarks. 

a 

0 

6.320 

590  99* 

584.674 

Bench  mark  on  monu- 

b 

150 

2.12 

588.87 

[ment  No.  51. 

c 

200 

6.38 

584.61 

TP.d 

3.561 

584.243J   .0.312 

580.682 

On  rock  50  ft.  N.E.  of  c 

e 

280 

!     1.20 

583.04 

T.P./ 

400 

10.617 

594.31'      0.543 

583.700 

On  rock. 

g 

475 

5.82 

588.50 

h 

500 

4.16 

590.16 

k 

584 

3.245 

591.072 

B.M.on  stump  oak  tree 

readings  are  frequently  taken  only  to  hundredths  on  the  benches 
and  turning  points  and  to  tenths  on  the  others.  The  final  ele- 
vation of  the  bench  mark  k  may  be  checked  thus: 

584.674+20.498  -  14.100  =  591.072 

in  which  20.498  is  the  sum  of  the  back  sights  on  the  benches 
and  turning-points  and  14.100  is  the  sum  of  the  fore  sights  on 
such  points.  (Art.  9.) 

When  levels  are  run  merely  to  find  the  difference  in  elevation 
of  two  points  a  and  k  (Fig.  38)  the  column  of  distances  is  not 
needed  in  the  notes,  and  there  are  no  intermediate  stations  &,  c, 
et  g,  h.  It  is  well,  even  in  such  cases,  to  fill  out  the  column  of 
height  of  instrument  in  the  field,  and  to  check  the  final  result 
in  the  manner  indicated  above.  The  main  note  book  is  always 
kept  by  the  leveler,  but  the  rodman  should  also  keep  a  book 
in  which  he  records  all  readings  on  benches  and  turning  points, 
finding  their  elevations  and  the  heights  of  instrument  so  as  to 
check  the  computations  of  the  leveler. 

Prob.  24.  Explain,  with  a  diagram,  why  it  is  that  precision 
in  levelling  is  promoted  by  setting  the  instrument  midway  be- 
tween the  turning  points. 


AKT.  25.    CONTOURS  AND  PROFILES. 

In  Art.  2  it  was  stated  that  the  dimensions  of  a  field  are  the 
horizontal  projections  of  the  actual  boundary  lines  and  that 


76  LEVELING  AND   TRIANGULATION. 

the  area  is  that  included  between  the  projections  of  the  bound- 
aries. It  is  evident  that  a  map  made  under  these  conditions, 
while  giving  a  clear  idea  of  the  shape  and  size  of  the  property, 
will  convey  no  information  as  to  the  character  of  the  surface, 
whether  high  and  uneven  or  flat  and  low.  These  distinctions 
would  be  evident  if  the  elevations  of  very  many  points  in  the 
field  were  written  at  the  proper  places  on  the  map,  but  so 
many  figures  would  render  other  features  of  the  map  indistinct, 
and  hence  another  plan  of  indicating  the  elevations  has  been 
adopted.  If  the  surface  of  the  ground  were  cut  by  a  series  of 
horizontal  planes  at  equal  distances  apart,  the  intersection  of 
each  plane  and  the  ground  would  be  an  irregular  line  connect- 
ing all  points  having  the  elevation  of  that  plane.  These  inter- 
sections called  Contour  Lines,  are  plotted  on  the  map  and  show 
at  a  glance  the  elevations  and  slopes  of  all  parts  of  the  fie^d 
with  a  precision  dependent  upon  the  nearness  of  the  planes  ta 
each  other.  A  clear  conception  of  the  utility  of  the  contoui 
lines  as  the  means  of  judging  of  the  features  of  a  surface  is 
formed  by  considering  the  surface  of  a  lake  as  the  intersecting 
plane.  The  shore  line  is  the  contour  having  the  elevation  ^f 
the  surface  of  the  lake;  if  the  water  were  to  fall  a  certain  dis- 
tance, the  horizontal  movement  of  the  shore  line  would  depend, 
not  only  upon  the  vertical  fall  of  the  surface  of  the  water,  but 
also  upon  the  declivity  of  the  ground,  being  small  where  the 
latter  is  steep  and  great  where  it  is  nearly  flat.  Hence  the 
slope  of  the  ground  is  judged  to  be  abrupt  wrhere  the  map 
shows  the  contour  lines  near  together,  while  the  slope  is  slight 
when  they  are  far  apart. 

The  position  of  the  contour  lines  is  not  generally  located  in 
the  field,  but  elevations  are  taken  at  points  where  the  slope  of 
the  ground  changes,  or  often  at  stakes  set  at  regular  intervals 
by  the  transit  and  chain.  These  elevations  are  then  plotted  in 
pencil  on  the  map  and  the  positions  of  points  at  the  elevation 
of  any  contour  are  found  by  interpolating  between  two  plotted 
elevations  one  of  which  is  above  and  one  below  the  required 
point;  the  contour  lines  are  then  drawn  by  connecting  points  of 
equal  elevation  by  a  curve;  the  elevation  of  the  contour  is 
-narked  on  it  and  the  plotted  figures  erased.  Let  the  field 


UONTuuKS   AND   PROFILES. 


7? 


ABCD,  Fig.  39,  be  divided  into  squares  100  feet  on  a  side  and 
elevations  taken  at  all  the  corners  as  shown,  and  let  it  be  re- 
quired to  locate  the  even  ten-foot  contours.  Beginning  at  any, 
as  the  upper  right-hand  corner,  the  ground  along  the  upper 
line  is  seen  to  fall  from  elevation  133  to  122  in  100  feet,  hence 
the  130  foot  contour  is  T3T  of  the  length  of  the  square  from  the 
corner,  and  the  120  foot  contour  is  seen  to  be  T2¥  of  the  distance 
from  the  second  corner  toward  the  third.  In  like  manner  all 
the  lines  are  gone  over  and  the  contours  are  then  sketched  in. 

110 116  1U 110  ^122 133 


FIG.  39. 

If  the  ground  is  very  uneven  many  complications  will  arise  in 
drawing  the  contours  from  the  plotted  elevations,  and  the  fol- 
lowing general  rules  will  be  useful  in  preventing  errors:  Con- 
tour lines  never  cross  each  other;  every  contour  on  one  side  of 
the  map  must  either  be  found  on  one  of  the  other  sides,  or  a 
second  time  on  the  first  one;  a  contour  not  crossing  any  side  of 
the  map  is  one  continuous  line,  returning  into  itself;  a  contour 
line  never  branches,  forming  a  loop ;  the  number  of  contours 
between  two  others  whose  elevations  are  alike  is  either  two, 
four,  or  some  other  even  numbev. 


78  LEVELING  AND  TRIANGULATIOH. 

The  intersection  of  the  surface  of  the  ground  by  a  vertical 
surface  is  called  the  Profile  along  that  line.  The  profile  is 
made  by  taking  the  elevations  at  known  intervals  along  the  de- 
sired course  with  the  level;  these  intervals  are  plotted  to  any 
suitable  scale,  and  at  each  point  where  an  elevation  was  taken 
an  ordinate  is  laid  off  whose  length  is  the  elevation  at  that 
point.  The  utility  of  the  profile  is  increased  by  making  the 
vertical  larger  than  the  horizontal  scale,  as  by  so  doing  the  rel- 
ative differences  in  elevation  are  made  much  more  apparent. 
The  profile  is  very  important  in  determining  the  grade  and  the 
probable  expense  of  building  streets,  railroads,  sewers  and 
drains.  In  the  case  of  a  street  profiles  of  the  middle  and  side 
lines  are  plotted  together,  using  ink  of  different  colors  if  neces- 
sary to  distinguish  the  three  lines,  and  the  suitable  position  for 
the  finished  grade  is  selected;  profiles  at  right  angles  to  the 
street  line,  or  cross-sections,  at  suitable  distances,  as  every  50 
feet,  are  plotted,  and  on  them  is  marked  the  position  of  the 
grade  line;  the  area  between  the  latter  and  the  surface  indi- 
cates the  amount  of  excavation  or  embankment  necessary. 

The  profile  of  any  line  on  a  contour  map  can  be  drawn  with- 
out any  additional  field  work,  since  the  elevations  of  the  inter- 
sections of  the  line  and  the  contours  are  known  from  the  height 
of  the  contours  themselves.  Thus  the  profile  of  a  line  through 
the  middle  of  the  upper  row  of  squares  in  Fig.  39  would  be 
made  by  first  drawing  the  line  in  pencil  across  the  map,  then 
the  elevation  at  the  right  end  is  130;  at  about  115  feet,  going 
toward  the  left,  the  elevation  is  120;  70  feet  further  110;  and  so 
on  across  the  map.  The  vertical  distances  on  a  profile  are 
usually  plotted  on  a  scale  from  5  to  20  times  as  large  as  the 
horizontal  scale. 

Prob.  25.  Draw  the  profiles  of  the  ground  along  the  lines 
AB  and  CD  in  Fig.  39,  making  the  vertical  scale  ten  times  the 
horizontal  scale.  Draw  also  the  profile  on  the  line  BG. 


ART.  26.    ADJUSTMENTS  OF  A  TRANSIT. 

The  adjustment  of  the  telescope  for  parallax,  described  in 
Art.  22,  must  be  made  every  time  it  is  used.     With  care  in 


ADJUSTMENTS   OF  A    filANSIT.  79 

handling  the  following  additional  adjustments  of  the  transit 
will  only  need  attention  at  rare  intervals,  but  the  instrument 
should  be  frequently  tested  to  see  if  it  is  in  order. 

Plate  Bubbles. — The  plane  of  each  s-mall  level  bubble  must 
be  parallel  to  the  horizontal  plate.  To  find  if  this  is  the  caso, 
carefully  level  the  instrument,  turn  the  alidade  through  about 
180  degrees,  and  note  whether  the  bubble  is  still  in  the  middle 
of  the  scale.  If  not,  move  the  capstan  screws  at  the  end  of  the 
bubble  tube  until  one  half  the  apparent  error  is  corrected. 
Then  level  the  instrument  again  and  repeat  the  operation. 
The  other  plate  bubble  is  adjusted  in  the  same  way. 

Collimation. — The  line  of  sight  must  be  perpendicular  to  the 
horizontal  axis  of  the  telescope.  To  find  if  this  is  the  case,  set 
up  the  transit  on  nearly  level  ground  and  sight  on  a  well-de- 
fined distant  object,  reverse  the  telescope  and  place  a  pin  about 
300  feet  from  the  instrument  in  the  opposite  direction ;  revolve 
the  alidade,  sight  to  the  same  object,  reverse  the  telescope,  and 
note  if  the  line  of  sight  strikes  the  pin.  If  not,  set  another  pin 
in  the  line  of  sight  by  the  side  of  the  first,  measure  the  distance 
between  them  and  place  a  third  pin  at  the  middle  of  that  dis- 
tance. Then  turn  the  capstan  screws  on  the  side  of  the  tele- 
scope until  the  vertical  cross-hair  has  moved  one  half  the  dis- 
tance from  the  second  to  the  third  pin.  Next  pull  up  all  the 
pins  and  repeat  the  operation  until  adjustment  is  secured. 

Horizontal  Axis. — The  horizontal  axis  of  the  transit  telescope 
must  be  parallel  to  the  horizontal  plate,  or  in  other  words  the 
standards  must  be  of  equal  height.  To  find  if  this  is  the  case, 
level  the  plate  bubbles,  elevate  the  telescope  as  high  as  prac- 
ticable and  sight  to  a  sharply  defined  object,  depress  the  tele- 
scope and  mark  a  point  on  the  ground  at  about  the  same  ele- 
vation as  the  instrument ;  then  reverse  the  telescope,  take 
another  sight  upon  the  same  object  and  mark  another  point  on 
the  ground.  If  these  points  do  not  coincide,  move  the  screws 
at  the  top  of  one  of  the  standards  until  the  vertical  hair  bisects 
the  distance  between  the  points.  Next  repeat  the  operation 
until  the  adjustment  is  perfect. 

Attached  Bubble. — The  attached  level  bubble  must  be  paral- 


80  LEVELING  AKD  TRIANGULATION^ 

lei  to  the  line  of  sight  of  the  telescope.  To  ascertain  if  this  is 
the  case,  set  up  the  instrument  and  level  the  telescope;  drive  a 
stake  A  about  a  foot  from  the  plumb-bob,  hold  a  level  rod  upon 
it,  and  take  the  rod  reading  ai  by  sighting  through  the  large 
end  of  the  telescope,  or  by  measuring  to  the  end  of  the  middle 
of  the  axis  of  the  telescope.  Drive  another  stake  B  about  400 
away  and  take  the  rod  reading  6j.  Next  set  the  instrument  as 
near  B  as  possible,  take  the  rod  reading  62  upon  it,  and  the 
rod  reading  az  upon  A.  Now  if  a\  —  bi  equals  «2  —  &a ,  the 
lines  of  sight  are  horizontal,  and  the  attached  bubble  is  in  ad- 


FIG.  40. 


justment.     If  not,  without  moving  the  level,  set  the  rod  on 
the  stake  A,  clamp  the  target  so'  that  the  rod  reads 


set  the  horizontal  cross-hair  on  the  target,  and  then  move  the 
bubble  into  the  middle  of  the  tube  by  the  screws  for  that  pur- 
pose at  the  end.  The  operation  is  then  to  be  repeated  until 
perfect  adj  ustment  is  secured.  This  is  called  the  peg  method 
of  adjustment. 

Vertical  Arc.  —  After  the  preceding  adjustments  are  made, 
the  vernier  of  the  vertical  arc  should  read  0°  00'  when  the  at- 
tached bubble  is  level.  If  this  is  not  the  case,  the  vernier  may 
be  moved  by  the  screws  at  its  ends  until  the  zero  points  coin- 
cide. This  adjustment  is  not  very  satisfactory,  and  instead  of 
making  it,  the  correction  may  be  noted  and  applied  to  each 
angle  when  it  is  read,  being  positive  for  angles  above  and 
negative  for  angles  below  the  horizontal  when  the  vernier  is 
too  far  toward  the  objective  end  of  the  telescope. 

Magnetic  Needle.  —  The  number  and  freedom  of  the  oscilla- 
tions of  the  needle  indicate  the  strength  of  its  magnetism.  If 
the  needle  becomes  sluggish  it  may  be  remagnetized  by  passing 
over  it,  toward  each  end,  the  pole  of  a  magnet  by  which  that 


COMPARISON   OF   TRANSITS.  81 

end  is  attracted,  returning  the  magnet  for  each  stroke  through 
a  circle  of  about  one  foot  diameter.  The  straightness  of  the 
needle  is  tested  by  reading  the  angle  between  the  two  ends, 
first  with  the  needle  is  its  normal  position,  then  when  turned 
end  for  end;  the  difference  is  double  the  real  error  and  the 
needle  should  be  bent  by  that  amount.  After  the  needle  has 
been  straightened,  the  two  ends  will  be  180°  apart,  if  the  pin 
upon  which  it  rests  is  in  the  center  of  the  circle.  If  this  is  not 
the  case,  clamp  the  instrument  in  any  position  and  bend  the 
pin  till  the  ends  of  the  needle  are  opposite  corresponding 
points;  then  turn  the  instrument  through  90°  and  again  make 
the  correction. 

Prob.  26.     Give  the  reasons  for  each  of  the  above  adjust- 
ments, drawing  a  figure  in  each  case. 


ART.  27.    COMPARISON  OF  TRANSITS. 

The  tests  of  the  telescope  and  its  attached  level,  described 
in  Art.  23,  may  be  applied  also  to  the  transit.  All  the  tests  of 
adjustments,  given  in  Art.  26,  should  likewise  be  made  upon 
a  transit  which  the  engineer  is  about  to  purchase.  In  addition 
to  these  there  are  others  relating  to  the  graduated  circle  which 
will  here  be  explained.  It  is  often  incorrectly  assumed  that 
the  larger  and  heavier  the  instrument  the  more  accurate  work 
it  is  capable  of  doing.  There  is  some  truth  in  this  with  res- 
pect to  the  level,  but  very  little  as  respects  the  transit.  For 
ordinary  work  a  transit  is  large  enough  if  it  has  a  circle  four 
inches  in  diameter.  Such  a  circle  can  be  made  to  read  to  half- 
minutes,  and  be  practically  as  easily  read  as  if  its  diameter 
were  six  inches.  Moreover,  the  extra  weight  of  the  larger  sizes 
does  not  materially  affect  the  stability  of  the  transit  as  that  is 
mainly  governed  by  the  stiffness  of  the  tripod  and  head.  For 
the  purposes  of  the  land  surveyor,  a  plain  transit, — that  is,  one 
without  attached  bubble  and  vertical  arc, — is  perhaps  sufficient. 
For  work  in  towns  and  cities  the  engineers'  transit,  which  has 
the  level  bubble  and  vertical  arc  and  also  two  verniers,  is  to  be 
preferred.  Unless  there  be  two  verniers  the  following  tests  of 
xhe  graduated  circle  cannot  be  made. 


LEVELING   AND   TRIANGULATION. 


Angular  Distance  of  Verniers. — The  angular  distance  be- 
tween the  zeros  of  the  two' verniers  should  be  exactly  180  de- 
grees, but  it  sometimes  varies  from  this  by  half  a  minute,  owing 
to  lack  of  care  by  the  maker.  To  ascertain  its  amount  the  ob- 
server must  be  able  to  estimate  halves  or  quarters  of  a  minute; 
this  is  not  difficult  if  the  two  lines  on  each  side  of  the  one  that 
apparently  coincides  are  also  regarded.  Vernier  A  is  set  ex- 
actly at  0°  and  then  the  amount  which  vernier  B  exceeds  or 
lacks  of  180°  is  read.  Next,  vernier  A  is  set  exactly  at  20°  and 
the  amount  wrhich  vernier  B  exceeds  or  lacks  of  200°  is  read. 
The  process  is  continued  at  intervals  of  twenty  degrees  ovel 
the  entire  circle,  and  the  results  are  tabulated  in  the  second 
and  fourth  columns  of  the  table  below,  the  plus  and  minus 
signs  denoting  the  excess  and  deficiency  of  the  supplement  of 
•the  angle  n  as  read  on  vernier  B.  The  table  is  so  arranged 
that  the  values  of  n  from  0°  to  180°  are  in  the  first  column, 
while  those  from  180°  to  360°  are  in  the  third  column,  and  the 
respective  discrepancies  for  the  two  parts  of  the  circle  are 
called  di  and  <72-  The  next  step  is  to  take  the  means  of  the 
corresponding  values  of  these  discrepancies,  observing  the 


^ 

j 

<i,+da 

d,  -  d* 

n 

»l 

«3 

2 

2 

0° 

-  45" 

180° 

.M5" 

0".0 

-  45.0 

20 

-  15 

200 

+  45 

+  15    .0 

-  30.0 

40 

-30 

220 

+  30 

0   .0 

-  30.0 

60 

00 

240 

+  45 

-1-22   .5 

-  22.5 

80 

-15 

260 

+  45 

+  15   .0 

-  30,0 

100 

00 

280 

+  30 

+  15    .0 

-  15.0 

120 

+  60 

300 

00 

+  30   .0 

+  30.0 

140 

+  60 

320 

-30 

+  15    .0 

+  45.0 

160 

+  60 

340 

—  45 

+    7   .5 

+  52.5 

D  =  +  120.0. 

algebraic  signs,  and  place  them  in  the  fifth  column.  The  sum 
of  these  is  D  —  +  120".0,  and  the  angular  distance  of  the  ver- 
niers is  180  degrees  plus  one-ninth  of  D,  or, 


Angular  distance  of  verniers  =  180° 


=  180°  00'  13", 


which  shows  that  an  error  of  13"  exists.     A  more  reliable  re- 
sult can  be  obtained  by  taking  readings  at  intervals  of  ten  do- 


COMPARISON   OF   TRANSITS.  «5 

grees  around  the  circle,  in  which  case  the  sum  D  is  to  be  divided 
by  eighteen. 

Eccentricity. — If  the  center  of  the  alidade,  to  which  the 
verniers  are  attached,  does  not  coincide  with  the  center  of  the 
graduated  plate,  it  will  revolve  around  the  latter  in  a  small 
circle.  When  the  vernier  is  on  a  line  joining  these  centers 
there  is  no  error,  but  for  any  other  position  all  the  readings  are 
affected  by  a  greater  or  less  error  of  eccentricity.  The  last 
column  in  the  above  table,  which  is  found  by  taking  the  means 
of  the  differences  of  the  two  sets  of  discrepancies,  shows 
roughly  the  errors  of  eccentricity.  From  it  there  appears  to 
be  no  error  when  vernier  A  reads  about  105°  or  285°,  and  a 
maximum  error  at  about  160°  or  340°.  A  closer  estimate  of 
these  quantities  can,  however,  be  made,  and  the  distance  be- 
tween the  two  centers  be  computed.  Let  each  of  the  quantities 
in  the  last  column  be  multiplied  by  the  sine  of  the  angle  in  the 
first  column  and  the  algebraic  sum  of  the  products  be  called  s. 
Let  each  quantity  be  also  multiplied  by  the  cosine  of  the  angle, 
and  the  algebraic  sum  of  the  products  be  called  t.  Using  only 
two  decimals  in  the  sines  and  cosines,  these  values  are  found 
to  be  s  =  —  20" .4  and  t  =  —  208".3.  Then  the  probable  angle 
HO  at  which  no  error  of  eccentricity  exists  is  found  by 

tan  n<>  -  -  -  =  _  10.2, 

8 

whence  n0  =  95^°.  Also  the  probable  maximum  value  of  the 
error  of  eccentricity  is,  if  m  be  the  number  of  readings  on  half 
the  circle, 

E= ^ =46".5. 

m  sin  n0 

Lastly,  the  radius  of  the  circle  in  which  the  center  of  the 
alidade  revolves  round  the  center  of  the  limb  is  to  be  found. 
Let  R  be  the  radius  of  the  graduated  limb,  which  in  this  case 
is  2£  inches;  then  the  radius  of  eccentricity  is 

r  =  %RE  sin  1"  =  0.00028  inches, 

which  is  the  distance  between  the  two  centers.  Although  this 
is  a  very  small  quantity,  it  yet  produces  sensible  errors  in  the 
readings. 

Bv  taking  several  sets  of  readings  in  the  manner  described 


84  LEVELING  AND  TRIANGULATION. 

a  fair  idea  can  be  obtained  of  the  angular  distance  between  the 
verniers  and  of  the  effect  of  eccentricity  on  readings  in  different 
parts  of  the  circle.  The  theory  of  errors  of  eccentricity  is  not 
given  here,  as  it  belongs  properly  to  higher  surveying,  but  it 
has  been  thought  well  to  explain  the  method  of  procedure  in 
order  to  enable  the  owner  of  a  transit  to  investigate  its  weak- 
nesses. It  fortunately  happens  that  in  precise  angle  measure- 
ments the  effect  of  these  sources  of  error  can  be  largely  elimi- 
nated by  the  method  of  repetitions  described  in  Art.  30. 

Prob.  27.  Test  two  transits  by  the  above  methods  and  write 
a  report  giving  the  observations  and  computations  in  full,  and 
comparing  the  two  instruments. 

ART.  28.    STANDARD  TAPES. 

In  town  and  city  surveying  linear  measurements  of  a  high 
degree  of  precision  are  often  necessary,  and  it  is  also  very  im- 
portant that  all  measures  should  be  referred  to  the  same  stand- 
ard. A  steel  tape  duly  certified  by  the  Bureau  of  Weights  and 
Measures  at  Washington,  is  the  most  convenient  standard,  and 
it  should  not  be  used  for  any  purpose  except  for  the  comparison 
of  other  tapes.  The  standard  tape  is  certified  to  be  correct  at 
a  given  temperature  when  under  a  given  pull ;  or  the  error  of 
its  length  is  stated  for  a  given  temperature  and  pull.  The  co- 
efficient of  expansion,  or  the  relative  change  in  length  for  one 
degree  Fahrenheit,  should  also  be  stated  in  order  to  render 
comparisons  at  other  temperatures  possible.  For  example,  a 
certain  tape  400  feet  long  is  stated  to  be  a  standard  at  56  de- 
grees Fahrenheit  when  under  a  pull  of  16  pounds,  and  its  co- 
efficient of  expansion  is  given  as  0.00000708.  At  a  temperature 
of  49  degrees  the  length  of  this  tape  will  be 

400  -  0.00000703  X  7  X  400  =  399.980  feet; 
at  a  temperature  of  70  degrees  its  length  will  be 

400  +  0.00000703  X  14  X  400  =  400.039  feet. 
To  compare  another  tape  with  the  standard  it  is  necessary  to 
know  its  coefficient  of  expansion  also.     In  order  to  determine 
this  the  tape  should  be  stretched  out  on  the  floor  of  a  large 


STANDARD   TAPES.  85 

room  whose  temperature  can  be  varied  or  be  kept  tolerably 
uniform.  With  a  spring  balance  at  each  end  it  is  pulled  to  the 
proper  tension,  the  thermometer  noted,  and  a  certain  length 
marked  on  two  tin  plates  temporarily  fastened  on  the  floor. 
The  temperature  is  then  raised  or  lowered,  and  the  operation 
again  repeated.  The  change  of  length  as  marked  on  the  tin 
plates  is  accurately  measured,  and  this  divided  by  the  total 
length  and  by  the  number  of  degrees  of  change  gives  the  co- 
efficient of  expansion.  For  example,  suppose  that  at  a  temper- 
ature of  41  degrees  a  length  of  60  feet  is  marked  off,  and  that 
this  is  done  again  at  a  temperature  of  79  degrees,  the  pull  be- 
ing the  same  in  both  cases,  and  the  change  in  length  being 
0.016  feet.  Then  the  coefficient  of  expansion  is 

(0.016  +  60)  +•  (79  -  41)  =  0.00000701. 

Owing  to  the  delicacy  of  this  operation,  a  single  result  is  not 
reliable,  and  hence  a  number  of  observations  should  be  made 
under  different  conditions  and  the  mean  of  the  various  results 
be  taken  for  the  final  coefficient. 

The  operation  of  comparing  a  tape  with  a  standard  consists 
in  laying  off  the  same  distance  by  both  and  thus  determining 
the  temperature  at  which  the  former  is  correct.  The  pull  on 
the  tape  may  be  selected  to  agree  with  its  size,  but  the  pull  on 
the  standard  must  always  be  the  given  assigned  pull.  As  an 
example,  let  the  standard  be  exactly  400  feet  long  at  56  degrees 
Fahrenheit  when  under  16  pounds  pull,  and  its  coefficient  of 
expansion  be  0.00000703.  Let  the  tape  to  be  tested  be  800  feet 
long,  its  coefficient  of  expansion  being  0.00000690.  With  the 
standard  300  feet  is  laid  off  with  the  pull  of  16  pounds,  and  the 
temperature  is  noted  as  63  degrees.  With  the  tape  300  feet  is 
also  laid  off  under  a  pull  of  18  pounds,  the  temperature  being 
noted  as  64  degrees.  The  second  distance  is  found  to  be  0.039 
feet  longer  than  the  first.  Now  let  t  be  the  temperature  at 
which  the  tape  is  correct  under  18  pounds  pull,  then 

800[1  +  0.00000690(64°  -  t)]  -  300[1  +  0.00000703(63°  -  56°)] 

=  0.039, 

from  which  t  is  found  to  be  38  degrees.  The  tape  is  therefore 
a  standard  at  38  degrees  Fahrenheit  when  under  18  pounds 


86  LEVELING  AND   TRIANGULATION. 

pull,  and  a  measurement  I  made  by  it  at  any  other  temperature 
have  the  true  value  Z  +  0.00000690(r—  38°y. 


If  the  tape  is  to  be  used  under  different  pulls  its  coefficient  of 
stretch,  or  relative  change  in  length  for  one  pound  pull,  should 
also  be  determined.  The  operation  for  doing  this  is  similar  to 
that  above  described  for  the  coefficient  of  expansion,  except 
that  the  temperature  should  be  constant  and  the  pull  be  varied. 
For  example,  let  a  length  of  300  feet  be  marked  off  at  15 
pounds  pull  and  again  at  19  pounds  pull,  and  let  the  change  in 
length  be  0.026  feet.  Then  the  coefficient  of  stretch  is 
(0.026  -5-  300)  -T-  (19  -  15)  =  0.0000216.  Any  length  I  made  un- 
der a  pull  P,  other  than  the  standard  pull  of  18  pounds,  will 
then  have  the  true  value  l-\-  0. 000021 6(P-  18)Z,  provided  the 
standard  temperature  of  38  degrees  exists. 

Sometimes  the  tape  is  stretched  over  two  supports  A  and  B, 
and  thus,  owing  to  the  sag,  the  measured  distance  is  too  long. 

Let  I  be  the  distance  read  on  the 
tape  under  a  pull  P,  let  d  be  the 

A  B        deflection  or  sag   at  the  middle, 

FIG.  41.  and  w  the  weight  of  the  tape  per 

linear  foot.     The  curve  of  the  tape  is  closely  that  of  a  parabola, 

and  if  L  be  the  horizontal  distance  L  =  I  —  -  — ,  very  nearly. 

o  L 

Also  taking  moments  at  the  middle  of  the  span  Pd  =  $wl .  ty. 
Eliminating  d  from  these  two  equations  the  adjusted  length  is 

found  L  =  I  —  fi(op-)  ^     For  example,  let  w  =  0.0066  pounds 

per  foot,  P  =  16  pounds,  and  I  =  309.851  feet,  then  L  = 
309.642  feet.  If  the  distance  AB  be  subdivided  into  n  equal 
spaces  by  stakes  whose  tops  are  on  the  same  level  as  those 

at  A  and  B,  then  L  =  I  —  5(-— •=}  I.     For  instance,  if  n  =  7, 
o  \&n.L  I 

Ahen  for  the  above  data  L  —  309.847  feet. 

To  recapitulate:  Let  t  be  the  temperature  and  p  the  pull  at 
which  a  tape  is  a  standard,  let  T  be  the  temperature  and  P  the 
pull  at  which  a  measurement  I  is  taken,  let  e  be  the  coefficient 
of  expansion  and  s  the  coefficient  of  stretch,  let  w  be  the 


BASE    LINES.  87 

weight  of  the  tape  per  linear  foot,  and  if  sag  exists  let  n  be  the 
number  of  equal  spaces  in  the  distance  I.  Then 

Correction  for  temperature  —  -j-  e(T  —  t}l', 
Correction  for  pull  =  -j-  s(P  —p)l\ 

Correction  for  sags  =  —  ^(—  -^)  I. 

i^4  \  ?ijt     / 

For  example,  let  t  =  56  degrees,  p-  16  pounds,  e  =  0.00000703, 
5  =  0.00001782,  M  =  0.0066  pounds  per  foot;  let  a  distance 
309.845  feet  be  measured  at  a  temperature  of  49£  degrees  under 
a  pull  of  20  pounds,  there  being  7  subdivisions  in  the  line. 
Then  the  correction  for  temperature  is  —  0.0142  feet,  that  for 
pull  -f  0.0221  feet,  and  that  for  sag  —  0.0028  feet.  The  ad- 
justed measured  distance  is  hence  309.850  feet. 

Lastly,  if  the  measurement  is  made  upon  a  slope  it  must  be 
reduced  to  the  horizontal  by  multiplying  it  by  the  cosine  of  the 
angle  of  slope.  It  is,  however,  generally  best  to  find  the 
difference  of  elevation  of  the  two  ends  of  the  line  by  leveling. 
If  A  be  this  difference  and  L  the  length  on  the  slope,  the  hori- 


zontal distance  is  ^JL?  —  A2.  For  instance,  if  the  length 
309.850  feet  has  2.813  feet  as  the  difference  of  level  of  the 
ends,  then  the  horizontal  distance  is  309.838  feet. 

Prob.  28.  A  tape  is  a  standard  at  41  degrees  Fahrenheit 
when  under  16  pounds  pull  and  no  sag,  its  coefficient  of  expan- 
sion  being  0.0000069  and  its  coefficient  of  stretch  0.000019. 
Find  the  pull  P  so  that  no  corrections  will  be  necessary  when 
measurements  are  made  at  a  temperature  of  38  degrees  and 
with  no  sags. 

AKT.  29.    BASE  LINES. 

A  triangulation  necessarily  starts  from  a  measured  base 
whose  length  must  be  known  with  precision  if  the  territory  to 
be  embraced  by  the  triangles  is  large.  A  long  steel  tape,  duly 
standardized,  is  the  best  instrument  for  making  the  measure- 
ment. The  base  line  should  be  divided  into  divisions,  each 
shorter  than  the  length  of  the  tape,  and  stout  posts  be  set  at 
the  ends  of  the  base  and  at  the  points  of  division.  On  these 
posts  are  placed  metallic  plugs,  each  Jiaving  drawn  upon  it  a 


88 


LEVELItfQ  AND  TRIANGULATIOtf. 


fine  line  at  right  angles  to  the  direction  of  the  base.  The  ele- 
vations of  these  plugs  should  be  carefully  determined.  Each 
division  is  then  subdivided  into  equal  parts  by  light  stakes  set 
in  line  and  on  grade,  the  distance  between  the  stakes  being 
fifty  feet  or  less.  Oa  each  stake  two  small  nails  may  be  placed 
to  keep  the  tape  in  position. 

The  measurement  should  be  done  upon  a  cloudy  day  with  lit- 
tle wind,  in  order  to  avoid  errors  due  to  change  in  temperature. 
The  tape  is  suspended  over  two  plugs  and  upon  the  interme- 
diate stakes  and  pulled  at  both  ends  by  spring  balances  to  the 
desired  tension.  At  one  plug  a  ten  foot  mark  on  the  tape  is 
made  to  coincide  with  the  fine  line  on  the  plug,  and  at  the 
other  end  a  mark  is  made  on  the  tape  directly  over  the  fine  line 
on  that  plug.  The  odd  distance  can  then  be  measured  with  a 
separate  scale  to  the  nearest  thousandth  of  a  foot.  Several 
measures  of  each  division  should  be  made  with  different  pulls, 
and  the  temperature  be  noted  at  each  reading. 

The  following  field  notes  of  a  short  base  measured  by  stu- 
dents of  Lehigh  University  will  illustrate  the  method  of  opera- 
tion. There  were  three  divisions,  designated  as  I,  II,  and  III, 


^8 

. 

Difference 

«J 

§ 

3 

CQ.2 

in 
Elevation 

g,g 

Pull. 

Observed 
Distance. 

Remarks. 

°> 

X'3 

of  Ends. 

c  s 

S 

fe 

EH 

feet 

pounds 

feet 

in 

7 

2.813 

51° 

16 

309.865 

Base  EG. 

50^ 
50^ 

18 
20 

309.857 

309.842 

Oct.  3,  1888,  P.M. 

50 

16 

309.870 

50 

18 

309.857 

Cloudy,  with 

ii 

7 

5.618 

48 

20 
16 

309.845 
332  .  746 

slight  wind. 

47| 

18 

332.727 

47? 

20 

332.712 

47 

16 

332.740 

47 

18 

332  726 

47 

20 

332  715 

i 

6 

7.924 

47 

16 

279.850 

47 

18 

279.843 

47 

20 

279  .  832 

48 

16 

279.848 

48i 

18 

279.840 

48 

20 

279.837 

the  first  having  six  and  the  others  seven  subdivisions.     The 
steel  tape  used  was  about  400  feet  long.    It  was  stated  by  the 


BASE    LINES. 


89 


makers  to  be  a  standard  at  56  degrees  Fahrenheit  when  under 
a  pull  of  16  pounds  and  having  no  sag.  By  a  series  of  experi- 
ments its  coefficient  of  expansion  had  been  determined  to  be 
0.00000703,  its  coefficient  of  stretch  0.00001782,  and  its  weight 
per  linear  foot  0.0066  pounds.  In  order  to  adjust  the  field  re- 
sults the  expressions  deduced  in  the  last  article  hence  are 

Correction  for  temperature  =  -  0.00000703  (56  —  T)l\ 
Correction  for  pull  =  +  0.00001782  (P  —  16)J; 


Correction  for  sag 


-  -  0.00001815  - 


from  which  the  corrections  are  computed.     For  example,  foi 
division  III.  where  n  =  7,  the  mean  of  the  observed  distances 


Temp 
T. 

Pull 
P. 

Observed 
Distance. 

Corrections. 

Adjusted 
Distance. 

Temp. 

Pull. 

Sag. 

51° 

50^ 

5oy2 

50 
50 
49^ 

Ibs. 
16 
18 
20 
16 
18 
20 

feet 
309.865 
.857 
.842 
.870 
.857 
309.845 

feet 
-  0.0109 
-  0.0120 
-  0.0120 
-  0.0131 
-  0.0131 
--  0.0142 

feet 
0 
+  0.0110 
+  0.0221 
0 
+  0.0110 
-f  0.0220 

feet 
-  0  0043 
-  0.0034 
-  0.0028 
-  0  0043 
-  0.0034 
-  0.0028 

feet 

309.850 
.853 
.849 
.853 
.8515 
309.850 

mean  =  309.856                                                       mean  =  309.851 
n  =  7       h  =     2.  813  feet             Final  horizontal  distance  =  309.838 

is  309.856  feet,  and  this  is  taken  as  the  value  of  I  in  all  cases. 
The  corrections  being  found,  the  adjusted  inclined  distances 
are  obtained,  and  their  mean  309.851  is  the  value  of  the  in- 
clined length.  Lastly,  this  is  reduced  to  the  horizontal,  giving 


j/309.8512  —  2.813'2  -  309.838  feet  as  the  final  result. 

Proceeding  in  the  same  manner  with  divisions  II  and  I  the 
corrections  are  found  and  the  sum  of  the  three  horizontal  dis- 
tances is  922.223  feet,  which  is  the  final  result  from  the  field 
work  above  given.  The  probable  uncertainty  of  this  result  is 
less  than  1  part  in  150,000,  which  shows  that  work  of  a  high 
degree  of  precision  can  be  done  with  a  steel  tape  whose  con- 
stants are  known. 

Prob.  29.  Compute  the  adjusted  inclined  lengths  and  the 
Snal  horizontal  lengths  of  divisions  II  and  I  of  the  above  has* 
line. 


90  LEVELING   AND   TRIANGULATION. 

ART.  30.     TRIANGULATION.   . 

The  process  of  triangulation,  after  tlie  base  is  measured, 
ionsists  in  observing  the  angles  of  all  the  triangles.  The  data 
are  thus  at  hand  for  computing  the  lengths  of  all  the  sides.  If 
the  azimuth  of  one  side  is  known,  or  has  been  obtained  by  the 
method  of  Art.  40,  the  azimuths  of  all  the  other  sides  are  easily 
found.  Lastly,  the  latitudes  and  longitudes  of  the  stations  of 
the  triangulation  are  computed  (Art.  8). 

In  triangulation  angle  measurements  are  required  to  have  a 
precision  greater  than  the  least  reading  of  the  vernier  will  give, 
and  the  method  of  repetitions  is  to  be  used.  To  illustrate  the 
principle  let  LOM  be  the  angle  to  be  measured.  Setting  the 
vernier  at  0°  00'  point  first  on  Lf  unclamp  the  alidade,  and 
point  on  M.  Now,  without  reading  the  vernier,  unclamp  the 
limb,  point  on  L,  unclamp  the  alidade,  and  point  on  M.  The 
vernier  has  thus  traveled  twice  over  the  arc,  and  if  it  be  now 
read  the  value  of  the  angle  is  one  half  the  reading.  If,  how- 
ever,  a  third  repetition  is  made  before  reading,  the  value  of  the 
angle  is  one  third  of  the  final  reading.  Thus  the  effect  of 
repeating  an  angle  is  to  divide  the  error  of  the  vernier  reading 
by  the  number  of  repetitions.  More  than  four  repetitions  are, 
however,  not  usually  advisable,  since  the  effort  of  clamping  is 
to  introduce  a  constant  tendency  to  error  in  one  diiection. 

The  process  of  repetition  in  any  important  case  should  be  so 
conducted  as  to  eliminate  the  effects  of  the  err>rs  of  non-adjust- 
ment, those  due  to  imperfections  of  the  graduated  limb,  and 
those  due  to  pointing  and  clamping.  Errors  due  to  lack  of 
level  of  the  limb  and  those  due  to  setting  the  instrument  or 
signals  in  the  wrong  position  cannot,  however,  be  eliminated, 
and  hence  great  care  should  be  taken  that  these  do  not  exist. 
Errors  due  to  collimation  and  to  the  horizontal  axis  of  the 
telescope  may  be  eliminated  by  taking  a  number  of  repetitions 
with  the  telescope  in  the  direct  position  and  an  equal  number 
with  it  in  the  reverse  position.  Errors  due  to  angular  distance 
between  the  verniers  and  to  eccentricity  of  the  graduated  limb 
may  be  eliminated  by  reading  both  verniers  and  taking  their 
mean.  Errors  due  to  inaccurate  graduation  may  be  eliminated 


91 


by  taking  readings  on  different  parts  of  the  circle.  Errors  due 
«o  pointing  and  clamping  may  be  largely  eliminated  by  taking 
one  half  of  the  repetitions  in  one  direction  and  the  other  half 
in  the  reverse  direction. 

The  following  form  of  field  notes  shows  four  sets  of  measure- 
ments of  an  angle  HOKt  each  set  having  three  repetitions. 
The  first  and  fourth  sets  are  taken  with  the  telescope  in  the 
direct  position,  the  second  and  third  with  it  reversed.  The 
first  and  second  sets  are  taken  by  pointing  first  at  H  and 
secondly  at  K,  the  third  and  fourth  are  taken  by  pointing  first 
at  K  and  secondly  at  H.  At  each  reading  both  verniers  are 
read.  The  vernier  is  never  set  at  zero,  but  the  reading  before 
beginning  the  set  is  taken,  this  being  made  to  differ  by  about 
90  degrees  in  the  different  sets  so  as  to  distribute  the  readings 
over  the  entire  graduation.  After  completing  a  repetition  both 
v  erniers  are  again  read.  In  the  first  and  second  sets  the  mean 
final  reading  minus  the  mean  initial  reading  is  divided  by  3, 
the  number  of  repetitions,  to  give  the  angle  as  determined  by 
t'aat  set.  In  the  third  and  fourth  sets  the  initial  reading  minus 
the  final  reading  is  divided  by  3.  If  very  accurate  work  is 
required  four  or  eight  additional  sets  may  be  taken  on  different 
parts  of  the  circle,  and  the  mean  of  all  will  be  the  probable 
value  of  the  angle. 


all 

1 

ft? 

Reading. 

Angle. 

.2  > 

w 

& 

| 

A 

B 

Mean 

Remarks. 

oog 

d 

'oS 

0           1 

•) 

p 

H 

Oft 

fc 

^ 

H 

8 

D 

20    04 

.00 

30 

15 

62    25    10 

Angle  at  station  O, 
Sept.  30,  1895,  3  p.m. 

K 

207    19 

30 

60 

45 

Brandis  Transit,  No.  716. 

H 

110    12 

30 

30 

30 

K 

3 

JS 

257    27 

60 

45 

52 

62    25    07 

John  Doe,  observer; 
R.  Roe,  recorder. 

K 

350    02 

00 

15 

07 

Air  hazy,  no  wind. 

3 

# 

62    25    33 

H 

162    43 

15 

30 

22 

K 

80    56 

15 

00 

08 

80  -f  360  =  440°. 

3 

D 

62    25    35 

H 

253    39 

00 

45 

22 

Mean  of  four  sets, 

HOK  =  62°  25'  21". 

| 

1 

92  LEVELING   AND   TRIANGULATION. 

In  repeating  angles  the  following  points  should  be  noted ; 
The  instrument  should  never  be  turned  on  its  vertical  axis  by 
taking  hold  of  the  telescope  or  of  any  part  of  the  alidade  ;  the 
limb  should  never  be  clamped  when  the  verniers  are  read  ;  the 
observer  should  not  walk  around  the  instrument  to  read  the 
verniers,  but  standing  where  the  light  is  favorable  he  should 
revolve  the  instrument  so  as  to  bring  vernier  A  and  then  vernier 
B  before  him  ;  the  observer  should  not  allow  his  knowledge  of 
the  reading  of  vernier  A  to  influence  him  in  taking  that  of  B ; 
care  must  be  taken  to  turn  the  clamps  slowly  and  not  too 
tightly.  If  these  precautions  be  taken  the  value  of  an  angle 

JOOOr 


"3000  4000  5000  6000 

FIG.  42. 

can  be  obtained  to  a  high  degree  of  precision  with  a  transit 
reading  only  to  minutes. 

The  stations  of  the  triangulation  should  be  points  which  are 
not  liable  to  be  lost,  such  as  holes  drilled  in  rocks  or  in  monu- 
ments firmly  planted  in  the  earth.  In  the  survey  of  a  town, 
however,  some  points  may  be  used  upon  which  the  transit  can- 
not be  set,  as  for  instance  church  spires,  but  these  must  be  so 
selected  that  they  can  be  seen  from  many  other  stations.  Care 
should  be  taken  that  all  the  triangles  are  well  proportioned, 
and  in  general  this  will  be  secured  when  no  angle  is  less  than 
30  degrees  or  over  150  degrees. 

A  triangulation  forms  the  framework  of  a  map.  All  its  sta- 
tions being  accurately  located,  a  traverse  may  start  at  any  one 
and  take  the  notes  necessary  for  a  map  of  that  vciniity,  check- 


TRIANGULATION. 


93 


ing  the  field  work,  perhaps,  by  ending  at  another  station.  Thus 
there  is  no  trouble  in  joining  different  surveys,  for  all  are  con- 
nected with  the  same  skeleton  framework.  In  plotting  the 
maps  a  coordinate  system  of  lines  1000  feet  apart  is  first  drawn 
and  upon  it  the  triangulation  stations  are  located  ;  from  these 
the  various  traverses  or  stadia  lines  are  laid  off  as  indicated  by 
the  field  notes.  The  precision  of  triangulation  work  will 
depend  upon  the  purpose  for  which  it  is  to  be  used ;  for  or- 
dinary town  or  topographical  surveys  it  will  perhaps  be  suf- 
ficient if  the  lengths  of  the  lines  and  the  coordinates  of  the 
stations  are  found  to  the  nearest  tenth  of  a  foot. 

In  Fig.  42  is  represented  a  small  triangulation  system  in 
which  EG  is  the  base  line  and  P  a  spire.  All  the  angles,  ex- 
cept those  at  P,  were  observed  by  the  method  of  repetitions, 
and  a  part  of  the  final  results  of  the  computations  are  given  in 
the  table  below.  Here,  as  in  Chapters  I  and  II,  the  azimuths 


Line. 

Azimuth. 

Distance, 
feet. 

Station- 

Latitude, 
feet. 

Longitude, 
feet. 

AQ 

186°  49'  38" 

404.57 

A 

2014.83 

3406.63 

AE 

>25   36  07 

778.95 

E 

2717.30 

3743.23 

AP 

,91    25  54 

593.55 

G 

2804  40 

4661.32 

E  A 

205    36  07 

778.95 

H 

2458.20 

5379.37 

EG 

84    34  48 

922.22 

K 

2250.76 

5733.05 

EP 

160    18  15 

761.87 

M 

1290.02 

5266.68 

G  P 

219   25  28 

1041.35 

N 

988.38 

4435.91 

GH 

115   44  28 

797.15 

Q 

1613.13 

3358.54 

HP 

251    37  29 

1453.48 

MP 

299    16  15 

1452.09 

are  counted  from  the  north  around  through  the  east,  south,  and 
west,  while  latitudes  are  positive  toward  the  north  and  longi- 
tudes positive  toward  the  east.  This  is  the  usual  method  in 
land  and  town  surveying.  It  should  be  said,  however,  that  in 
geodetic  work  and  in  extended  topographical  surveys  the 
azimuths  are  often  counted  from  the  south  around  through  the 
west,  north,  and  east,  while  latitudes  are  taken  as  positive 
toward  the  north  and  longitudes  as  positive  toward  the  west. 

Prob.  30.     Compute  the  latitude  and  longitude  of  P  from 
the  above  data  by  several  different  methods. 


94  TOPOGRAPHIC  SURVEYING. 

\ 

CHAPTER    IV. 

TOPOGRAPHIC  SURVEYING. 

ART.  31.    LARGE-SCALE  TOPOGRAPHY. 

THE  scale  to  which  topographic  maps  are  drawn  depends 
upon  the  use  for  which  they  are  designed  ;  if  it  is  desired  to 
show  a  large  extent  of  territory  at  once,  the  scale  will  be  de- 
termined by  the  size  of  the  finished  map  which  will  be  most 
convenient  for  use  ;  on  the  other  hand,  if  it  is  desired  to  show 
a  smaller  territory  but  with  more  minuteness,  a  larger  scale 
could  be  adapted  to  the  same  size  sheet  as  before.  The  scale 
of  the  map  influences  the  degree  of  accuracy  employed  in  the 
field  work  and  also  the  appearance  of  the  signs  used  in  repre- 
senting the  various  topographic  features. 

Under  the  term  large  scale,  it  is  intended  to  include  maps 
plotted  to  a  scale  larger  than  400  feet  to  an  inch.  Such  maps 
are  designed  to  show  the  contour  lines  with  from  2  feet  to  10 
feet  intervals,  the  former  distance  being  applicable  in  case  the 
country  is  flat,  and  the  latter  where  the  slopes  are  abrupt  or 
where  less  precision  is  required.  All  roads  and  streets, 
whether  highways  or  on  private  property,  are  shown  and  also 
the  positions  of  the  property  lines.  Dwellings  and  other 
buildings  are  represented  in  their  true  shape  and  with  dimen- 
sions drawn  to  the  scale  of  the  map.  The  positions  of  isolated 
trees  are  located  by  measurement,  as  are  also  the  boundaries  of 
woods.  If  a  stream  is  to  be  shown,  both  sides,  instead  of  the 
middle  line  alone,  are  plotted  unless  the  width  is  so  small  that 
one  stroke  of  the  pen  would  cover  both  sides.  It  sometimes 
happens  that  objects  have  to  be  plotted  out  of  proportion  to 
the  rest  of  the  map  because,  mechanically,  it  is  impossible  to 
represent  them  on  the  proper  scale.  It  is  quite  impracticable 
to  plot,  or  for  the  eye  to  distinguish,  distances  on  the  map  of 
less  than  yj-^  of  an  inch  ;  if  the  scale  of  the  map  is  200  feet  to 
an  inch,  -^  of  an  inch  represents  2  feet  and  hence  objects  of 
less  size  than  that  are  indicated  by  one  line.  A  specimen  of  a 
large-scale  topographic  map  is  shown  in  Fig.  43, 


LARGE-SCALE   TOPOGRAPHY. 


95 


Fio.  43. 


96  TOPOGRAPHIC   SURVEYING. 

The  conventional  signs  used  in  illustrating  topographic 
characteristics,  whether  indicating  the  nature  of  the  ground  or 
of  the  crops  growing  upon  it,  are  designed  to  bear  some  degree 
of  resemblance  to  the  objects  they  are  to  represent ;  the  motive 
in  the  use  of  the  signs,  however,  is  to  convey  information  con- 
cerning the  character  rather  than  the  actual  appearance  of  the 
objects,  and  hence  no  attempt  is  made  to  draw  the  signs  to  the 
scale  of  the  map,  other  than  to  make  them  of  such  size  and 
weight  as  will  harmonize  with  the  other  parts  of  the  drawing. 
It  is  of  the  first  importance  that  the  topographic  drafts- 
man be  entirely  familiar  with  the  exact  appearance  of  the 
signs  he  wishes  to  use  ;  especially  is  this  true  if  the  drawing 
is  to  be  on  a  large  scale  where  no  marks  are  made  at  random, 
but  each  one  is  to  perform  a  definite  part  in  producing  the 
general  effect  of  the  whole.  Some  of  the  signs  in  most  fre- 
quent use  are  shown  in  the  sketches  given  in  Fig.  44. 

Care  must  be  taken  that  the  signs  are  so  made  as  to  avoid  A 
flat  appearance,  which  is  a  common  fault  of  otherwise  well  ex- 
ecuted drawings.  It  is  a  universal  custom  to  consider  the 
light  as  coming  from  the  direction  of  the  upper  left-hand  cor- 
ner, in  which  case  the  shadow  will  be  on  the  lower  and  right- 
hand  sides  of  the  figures,  and  accordingly  those  parts  are  made 
with  a  somewhat  heavier  stroke.  In  making  the  signs  for 
grass  the  shade  is  very  slight,  except  in  swamps  where  the 
shadow  is  drawn  under  each  tuft,  but  in  case  of  the  forest  it 
is  of  great  importance  in  relieving  the  appearance  of  sameness 
which  the  map  would  otherwise  have.  In  representing  water 
and  the  shore,  it  is  a  common  fault  to  make  the  line  of  the 
latter  too  light,  the  distinction  between  this  line  and  the  first 
shade  line  of  the  water  should  be  very  marked. 

Scales  are  frequently  designated  as  ratios ;  thus  a  scale  of 
-gjpj^  is  such  that  any  actual  line  in  the  field  is  25,000  times 
as  long  as  its  representation  on  the  map.  A  scale  of  400  feet 
to  an  inch  is  the  same  as  4800  inches  to  an  inch,  or  ^y^ 
as  commonly  expressed. 

Prob.  31.  How  many  feet  are  represented  by  one  inch  on  a 
tcale  of  j^Vtf?  How  many  acres  are  represented  by  one  square 
inch  on  a  scale  of 


LAKGE-SCALE  TOPOGRAPHY. 


97 


,  ..Round  Leaf 

,p iu> ;. y. •  <io ,f  «  r /•> v,  /yo¥3     <^' '^-'^ -> "*-i«KV x ' \\\\\ 
V  CP  £,V^>' > ;       '• ' '  ov^S  •    r^<  C  ulti vated  Land x 


o    o    o    o    o    o    o 

•   »   •      Cotton  -    •  •    • 

»  0    e   o   o   o   o  o 
o    o    o    o    o    o    o 


*sHj5z-  "^  "  ~  *.  —  -^    •"•  Vvx"v::>     I 

.-•T~~        *^ix/— -r~~—  "2£fc.\''.*.r  •*»•' 

~     •«tf!/'j?;_        ""  "".."^    -     •  '  '     ~      *.     •'..''.;.    •      \L 

S  »•".:•«   \^~ 


*    *    * 


o.  44. 


98  TOPOGRAPHIC   SURVEYING. 


ART.  32.    SMALL-SCALE  TOPOGRAPHY. 

In  surveys  covering  very  large  areas  the  details  are  made 
subordinate  to  the  general  features  of  the  country.  In  the 
previous  article  several  reasons  for  so  doing  were  stated,  and 
in  addition,  the  usefulness  of  the  maps  is  not  such  as  to  war- 
rant so  great  expenditure  as  would  be  involved  in  making  the 
maps  to  a  large  scale.  The  saving  in  the  cost  is  due,  partly  to 
the  fact  that  less  labor  is  necessary  in  plotting  the  maps,  but 
more  especially  to  the  economy  of  time  possible  in  making  the 
survey,  since  objects  need  be  located  with  only  such  precision 
as  will  make  the  errors  on  the  map  unobservable.  The  smaller 
the  scale  the  less  frequent  will  be  the  revisions  necessary  to 
keep  the  maps  reliable  since  the  objects  subject  to  change  are, 
for  the  most  part,  omitted  on  the  small-scale  maps. 

The  topographic  maps  made  by  the  United  States  Coast  and 
Geodetic  Survey  and  by  the  United  States  Geological  Survey 
are  drawn  to  the  scale  of  1  to  62,500,  1  to  125,000,  or  1  to  250,- 
000,  with  corresponding  contour  intervals  of  5  to  50  feet,  10  to 
100  feet  and  200  to  250  feet.  These  scales  are  seen  to  be  ap- 
proximately one,  two,  or  four  miles  to  the  inch.  The  largest 
scales  are  used  where  the  country  is  most  densely  populated 
or  where  it  is  flattest.  Some  small-scale  maps  show  the 
streams,  the  state,  county,  and  town  divisions,  the  highways, 
railroads,  and  canals  ;  but  private  ways  and  property  lines  are 
not  represented  ;  features  of  public  importance  being  given, 
and  those  of  a  temporary  nature  omitted. 

The  conventional  signs  used  on  the  small-scale  maps  are 
made  to  present  approximately  the  appearance  of  those  of 
larger  sca\es  when  seen  from  a  distance  ;  the  details  can  hardly 
be  distinguished  without  the  aid  of  a  magnifying  glaas. 
Buildings  are  represented  simply  by  black  rectangles  without 
much  regard  to  the  shape  or  size  of  the  houses  themselves. 
Isolated  trees,  small  orchards,  and  groves  are  not  shown,  but 
the  boundaries  of  forests  are  plotted  to  scale  and  the  interior 
is  filled  in  as  shown  in  Fig.  45,  with  signs  similar  to  those 
given  in  Fig.  44,  but  very  much  smaller.  The  highways  are 


SMALL-SCALE   TOPOGRAPHY. 


99 


FIG.  45. 


100  TOPOGRAPHIC  SURVEYING. 

represented  by  parallel  lines  of  uniform  distance  apart,  with- 
out regard  to  the  actual  width  of  the  road.  The  scale  of  Fig. 
45  is  f-fau,  while  that  of  Fig.  53  is  ^Tnr>  both  being  taken 
from  the  maps  of  the  Coast  and  Geodetic  Survey. 

The  use  of  colors  is  not  as  frequent  as  formerly,  but  the 
appearance  of  any  map  is  improved  and  its  utility  increased 
by  the  contrast  thus  made,  if  the  land  be  covered  with  a  light 
wash  of  burnt  sienna  with  the  contour  lines  of  a  darker  shade 
of  the  same  color,  and  the  water  colored  blue  ;  all  other  marks 
are  in  black. 

Prob.  32.  Draw  a  profile  of  the  surface  as  cut  out  by  a 
vertical  plane  through  the  NE  and  SW  corners  of  Fig.  45. 

ART.  33.    THEORY  OF  THE  STADIA. 

The  fundamental  principle  of  stadia  measurements  is  that 
of  similarity  of  triangles.  In  Fig.  46  let  T  represent  a  tube 
having  three  horizontal  hairs  and  let  vertical  graduated  rods 
be  held  in  the  positions  AB  and  AiBi.  The  eye  being  at  the 
end  E,  the  distances  (7Z£and  CiEoi  the  rod  from  E  are  directly 


FIG.  46. 

proportional  to  the  spaces  AB  and  AiBi  apparently  inter- 
cepted on  the  rods  by  the  cross-hairs.  This  simple  proportion 
is  modified  somewhat  in  practice  by  the  fact  that  a  telescope 
replaces  the  plain  tube. 

In  Fig.  47,  the  cross-hairs  are  at  a  and  &,  and  i  is  the  dis- 
tance between  them.  Rays  of  light  supposed  to  pass  outward 
from  a  and  b  are,  by  refraction  of  the  object  glass,  made  to 
intersect  at  0,  at  a  distance  from  the  lens  equal  to  the  focal 
length  of  the  telescope  ;  these  rays  intersect  the  rod  at  A  and 
B,  the  points  upon  which  the  hairs  a  and  b  are  apparently 
projected  by  the  eye  at  E.  If  the  rod  is  moved  to  any  other 


THEORY   OF   THE   STADIA.  101 

point  distant  d  '  from  0  the  space  intercepted  on  the  rod  by  the 
cross -hairs  will  have  the  same  relation  to  AB  that  d '  does  to 
d,  because  of  the  similarity  of  triangles  as  in  Fig.  46.  The 
total  distance  from  the  instrument  to  the  rod  is  D  =  <?+/+  d\ 
in  which  c  is  the  distance  from  the  plumb-bob  to  the  object 
glass  and  F  is  the  focal  length  of  the  telescope.  From  the 
figure  it  is  seen  that 

d:AB  ::/  :  i,       or      d  =  R£; 

i 

hence  D  =  (c  +/)  -f  R~ 

From  this  equation  it  would  appear  that  the  determination  of 
D  depends  upon  very  careful  measurements  of  f  and  i,  but 


FIG.  47. 


such  measurements  are  impracticable  and  unnecessary  since 

the  value  of  4  can  be  determined  by  trial  when  c  and  /  are 

i 

approximately  known.  The  distance  c  is  found  by  measuring 
from  the  axis  of  the  telescope  to  the  middle  .of  the  object, 
glass  when  the  telescope  is  focused  for  a  d^tii/^of  .about  390 
feet  or  a  mean  of  all  the  distances. that t are,  to  bs -measured. 
When  the  telescope  is  focused  for  an  infinite  ••ISst-unce  /  is  fbe 
space  between  the  object  glass  and  the  cross-hairs  ;  this  can 
readily  be  measured  with  sufficient  accuracy  when  the  focus  is 

for  an  object  a  mile  or  so  distant.     To  find  the  value  of  "4-, 

measure  from  the  center  of  the  instrument  any  convenient 
distance,  as  (c  -\-f)  -f-  200  feet,  along  level  ground  and  hold 
the  rod  on  the  point  thus  found.  Sight  to  the  rod  and  count 
the  number  of  spaces  on  it  between  the  upper  and  lower  hairs, 

then  the  constant  number  -V  can  be  found  from  the  equation 


10&  TOPOGKAPHIC   SURVEYING. 

D  =  (c  +/)  +  #£-.     Thus  let  c  =  5  inches,  /  =  7  inches,  the 

measured  distance  to  the  rod  201  feet,  and  the  space  intercepted 
on  the  rod  2.02  feet  ;  then 

201  =  (0.48  +  0.52)  +  2.02  4, 


This  would  be  a  very  awkward  factor  to  use  and  hence  it  is 
desirable  to  either  change  the  value  of  i  by  moving  the  hori- 

zontal hairs,  or  to  substitute  another  rod  on  which  the  gradua- 

f 
tions  are  of  such  size  that    ,-  multiplied  by  one  of  the   unito 

will  equal  100. 

To  adjust  the  hairs  to  fit  the  rod,  measure,  on  nearly  level 
ground,  some  convenient  distance,  as(c+/)  +  200  feet  from 
the  plumb-bob,  and  sight  upon  the  rod  held  at  that  distance 
from  the  instrument  ;  move  the  upper  hair,  by  means  of  the 
capstan  screw  for  the  purpose,  till  one  space  is  intercepted  on 
the  rod  between  the  upper  and  middle  hairs,  then  similarly 
apply  the  correction  to  the  lower  hair.  In  case  an  ordinary 
self-reading  level  rod  is  used  the  cross-hairs  would  intercept 
two  feet  on  it  when  the  distance  from  the  instrument  i?i 
(c+/)  +  200  feet. 

If  the  cross-hairs  are  fixed,  the  rod  can  be  so  graduated  that 
the  number  of  spaces  intercepted  on  it  by  the  hairs  will 
always  be  the  nu  diver  of  hundred  feet  that  the  rod  is  from  a 
point  (c-j-jO  feet  in  front  of  the  instrument.  Sight  to  the 
plain  rod.'  held  at  a  distance,  say,  (c  -\-f)  -{-  300  feet  from  the 
instrument  and  mark  where  the  upper  and  lower  hairs  inter- 
sect the  rod  ;  this  space  divided,  in  this  case,  by  three  is  then 
the  unit  by  which  the  whole  rod  is  to  be  graduated.  After  the 
units  are  marked  on  the  rod  they  are  sub-divided  into  ten  or 
twenty  equal  parts  to  aid  the  eye  in  estimating  distances  othei 
than  the  even  hundreds. 

When  the  rod  is  to  be  used  in  surveys  which  are  to  be 
plotted  to  a  small  scale,  the  constant  (c  +/)  is  often  disre- 
garded and  the  rod  is  graduated  accordingly.  The  rod  is  held 
at  distance  from  the  plumb-bob  which  is  supposed  to  be  about 


THEORY   OF   THE   STADIA. 


103 


a  mean  of  all  distances  to  be  measured,  and  so  graduated  that 
the  rod  reading  will  correctly  indicate  that  particular  distance. 
When  the  rod  is  held  nearer  the  instrument  the  indicated  dis- 
tance is  a  little  too  small  while  distances  greater  than  the 
mean  are  slightly  too  large.  If  the  rod  is  graduated  for  500 
feet  the  maximum  error  for  distances  between  100  feet  and 
1000  feet  will  be  about  1  foot. 

If  the  rod  is  to  be  always  used  in  open  country  where  the 
whole  of  it  can  be  seen  the  following  method  of  graduation 
may  be  adopted.  Hold  the  rod  at  100  feet  from  the  instrument 
and  mark  the  space  intercepted  by  the  cross-hairs,  the  upper 
one  being  sighted  to  the  uppermost  mark  on  the  rod  or  the 
lower  one  to  the  lowest  mark  ;  next  hold  the  rod  at  200  feet 
from  the  instrument,  direct  the  same  hair  as  before  to  the 
mark  at  the  end  of  the  rod  and  note  the  point  intersected  by 
the  other  hair.  The  graduations  for  the  entire  rod  are  made  in 
a  similar  manner  by  marking  the  spaces  actually  intercepted 
at  each  successive  100  feet  distance  from  the  instrument,  one 
hair  always  being  on  the  beginning  of  the  graduations. 

When  the  line  of  sight  is  inclined  to  the  horizontal  it  is 
evident  that  the  distance  indicated  on  the  rod  is  not  the  re- 
quired horizontal  distance  from  the  instrument.  If  the  rod  is 
held  perpendicular  to  the  line  of  sight,  the  reading  will  indi- 
cate the  inclined  distance  from  the  instrument  to  it ;  the  hori- 


FIG.  48. 


zontal  distance  can  then  be  found  if  the  angle  between  the 
line  of  sight  and  the  horizontal  is  known.  In  practice  it  is 
found  to  be  impracticable  to  hold  the  rod  at  right  angles  to 
the  line  of  sight ;  it  is  hence  placed  vertical  and  an  expression 
is  found  by  which  the  horizontal  distance  is  computed  from 
the  rod  reading  and  the  measured  vertical  angle  v 


104  TOPOGEAPHIC   SURVEYING. 

In  Fig.  48,  AB  is  the  reading  on  the  vertical  rod  and  A'B1 
that  when  the  rod  is  perpendicular  to  the  line  of  sight.  Since 
the  angle  AOB  is  small,  no  appreciable  error  will  result  if 
A  A'B  is  considered  as  90°;  then 

A'B'  —  AB  cos  v. 
A'B'  indicates  the  distance  OP,  and  TP  =  c  +f+  OP. 

T8=  TP  cos  v  =  (c  +f  +  AB  cos  v)  cos  0; 
D  —  (c  +/)  cos  v  +  R  cos2  v, 

when  12  is  the  distance  indicated  by  the  rod  reading.  The 
term  (c  -\-  /)  cos  v  may  always  be  taken  as  one  foot  without 
any  practical  error. 

The  difference  in  elevation  H  is  found  by  sighting  the 
middle  cross-hair  to  a  point  on  the  rod  at  the  same  height  a 
above  the  ground  that  the  telescope  is,  and  observing  the  ver- 
tical angle  v.  Thus, 

PS  —  TP  sin  v  —  (c  +/+  AB  cos  v)  sin  ®; 
or, 

H  =  (  c  +  /)  sin  v  -f-  R  sin  v  cos  v. 

For  values  of  v  less  than  4  degrees  the  terms  (c  +/)  sin  v  may 
be  neglected,  and  (c  +/)  may  generally  be  taken  as  one  foot. 

Prob.  33.  Let  (c  +/)  =  0.87  feet,  R  =  465  feet,  and  v  =3°, 
32'.  Compute  the  horizontal  distance  D  and  the  difference  in 
elevation  H.  What  error  results  if  (c  +/)  is  not  considered? 

ART.  34.     STADIA  REDUCTIONS. 

The  formulas  for  D  and  Ht  deduced  in  the  last  article,  involve 
much  labor  in  computation,  and  hence  Table  X  is  given  to 
facilitate  the  reductions.  As  an  example  of  its  use,  suppose 
that  (c  +/)  f°r  tue  instrument  is  1  foot,  and  that  a  certain  rod 
reading  gives  680  feet  for  a  vertical  angle  of  5°  26'.  Then, 
by  the  help  of  the  table, 

D  =  0.99  +  6.8  X  99.10  =  674.9  feet, 
H  =0.09  +  6.8  X    9.43=    64.2  feet, 

or,  D  =  674  feet  and  #=64.1  feet  if  the  value  of  (c  +/)  is 
not  taken  into  account. 

The  work  of  reducing  to  horizontal  distances  and  differences 


STADIA   REDUCTIONS. 


105 


of  elevation  the  results  of  a  single  day's  work  in  the  field  with 
the  stadia  is  exceedingly  tedious,  even  with  the  aid  of  Table  X, 
and  many  schemes  designed  to  lighten  this  labor  have  been 
suggested.  Of  these  devices  the  most  common  are  in  the  form 
of  diagrams  or  of  the  slide  rule.  The  objection  to  diagrams  is 
that  lines  crossing  at  very  acute  angles  have  an  indefinite  inter- 
section and  separate  diagrams  have  to  be  constructed  for,  at 
most,  every  ten  degrees  of  vertical  angle  and  also  separate  ones 
for  horizontal  distances  and  differences  of  elevation.  The  slide 
rule  performs  the  operations  with  considerable  accuracy  and 
dispatch,  but  the  cost  of  such  an  instrument  prohibits  its  use  in 
many  instances. 

In  Fig.  49  is  shown  a  sketch  of  an  apparatus  whose  efficiency 
has  been  tested  by  several  years'  use  and  which  may  be  made 


FIG.  49. 

by  any  student  of  average  manual  skill.  The  apparatus  con- 
sists  of  a  large  sheet  of  heavy  paper,  a  movable  wooden  arm, 
and  a  triangle.  Along  the  lower  edge  of  the  paper  is  a  gradu- 
ation to  some  convenient  scale  of  equal  parts  and,  about  the 
zero  of  this  as  a  centre,  an  arc  of  a  circle  is  drawn  through  or 
near  the  other  end  and  divided  into  degrees.  The  movable  arm 
and  the  longer  of  the  two  perpendicular  sides  of  the  triangle 
are  graduated  to  the  same  scale  as  that  on  the  paper. 
In  making  the  reduction  the  movable  arm  is  set  to  correspond 


106  TOPOGRAPHIC   SURVEYING. 

with  the  angle  of  elevation  or  depression,  V,  as  indicated  by 
the  circular  arc.  The  triangle  is  then  placed  as  shown,  so  that 
it  crosses  the  lower  scale  at  the  rod-reading  on  the  latter.  Since 
AB  is  perpendicular  to  OB  the  reading  on  the  scale  of  the  arm 
will  be  R  cos  V.  The  triangle  is  then  moved  into  the  position 
shown  by  the  dotted  lines  where  the  reading  on  the  horizontal 
scale  at  Bt  is  the  same  as  was  noted  at  B  or  R  cos  F.  With 
the  triangle  in  this  position  the  horizontal  distance  R  cos'  F 
may  be  read  at  C  on  the  scale  of  the  arm  and  the  difference  in 
elevations  at  Bt  on  the  scale  of  the  triangle.  The  constants  for 
the  instrument  must  be  added  to  these  results.  Since  BC  is 
small,  usually  less  than  an  inch,  the  operation  consists  prac- 
tically of  one  setting  for  the  two  reductions.  The  reductions 
for  the  transit  stations  should  always  be  checked  by  the  tables. 
As  an  example  of  reduction  let  F  be  22°  30'  and  R  be  200 
feet.  The  arm  is  set  at  22°  30',  as  shown  in  Fig.  49,  and  the 
triangle  is  so  placed  as  to  intersect  the  lower  scale  at  the  200 
mark.  The  reading  on  the  arm  is  seen  to  be  about  185,  so  the 
triangle  is  slipped  back  till  it  crosses  the  lower  scale  at  185. 
The  reading  then  at  C  is  about  171  and  at  B}  on  the  triangle  is 
slightly  over  70.  The  horizontal  distance  and  difference  of 
elevation  are  respectively  171  feet  and  70  feet  plus  corrections 
for  instrumental  constants. 

The  accuracy  of  the  above  example  does  not  of  course  com- 
pare with  that  possible  with  a  full-size  apparatus.  The  par- 
ticular one  described  has  an  arc  of  40  inches  radius  divided 
into  5-minute  spaces,  which  are  large  enough  to  make  readings 
to  single  minutes  practicable.  The  other  divisions  are  on  the 
scale  of  10  feet  to  the  inch,  so  that  tenths  of  a  foot  may  be 
easily  read.  The  apparatus  was  constructed  at  an  expense  of 
less  than  one  dollar,  and  with  it  from  140  to  150  reductions  per 
hour  have  been  made.  It  is  better  for  permanent  use  to  make 
the  graduations  on  a  drawing-board  instead  of  on  paper,  as  the 
latter  is  liable  to  shrink  or  expand  with  changes  of  temperature. 

Prob.  34.  Construct  an  apparatus  for  stadia  reductions  like 
that  above  described,  and  compare  the  precision  of  its  work 
mth  that  of  Table  X. 


FIELD   WORK.  10? 


ART.  35.     FIELD  WORK. 

The  topographic  survey  of  a  large  territory  is  preferably 
based  upon  a  system  of  triangulation,  which  will  afford 
numerous  checks  upon  the  stadia  traverses.  The  stations 
should  be  located,  not  only  to  secure  well-conditioned  triangles, 
but  also  so  that  they  may  be  of  the  greatest  use  to  the  topogra- 
phers. In  a  flat  wooded  country  a  triangulation  system  is 
carried  on  only  at  great  expense  of  erecting  towers,  and  in  such 
cases  it  is  sometimes  advisable  to  locate  the  permanent  refer- 
ence stations  by  means  of  carefully  conducted  traverses.  By 
whatever  method  they  are  established,  the  stations  should  be 
near  enough  together  to  furnish  means  of  verifying,  each  day, 
the  work  of  the  topographical  parties.  The  elevations  of  the 
stations  are  to  be  determined  and  other  bench  marks  estab- 
lished at  proper  intervals  by  precise  leveling,  in  order  that  the 
errors  arising  from  the  use  of  the  stadia  in  determining  heights 
may  be  confined  to  the  short  traverse  lines  between  the  princi- 
pal stations. 

The  transit  used  in  stadia  surveying  need  not  be  of  large 
size,  but  there  are  some  features  that  are  especially  essential 
in  instruments  for  this  purpose.  The  telescope  should  have  a 
perfectly  flat  field  of  view,  since  the  lines  of  sight  do  not  coin- 
cide with  the  optical  axis;  this  defect  furnishes  the  opponents 
to  the  use  of  the  stadia  with  their  strongest  argument.  The 
vertical  arc  should  be  of  superior  quality,  the  graduations 
being  upon  solid  silver,  and  there  should  be  means  of  adjust- 
ing the  vernier  so  that  the  reading  shall  be  zero  when  the  tele- 
scope is  level.  A  telescope  having  fixed  stadia  hairs  gives 
the  best  results,  but  can,  of  course,  be  used  only  with  a 
specially  prepared  rod.  The  horizontal  circle  should  have  its 
graduations  numbered  continuously  from  0*  to  360°  in  the 
direction  that  azimuth  is  reckoned,  and  there  should  be  means 
of  setting  off  the  magnetic  declination  so  that  the  needle  may 
indicate  north  or  south  when  the  line  of  sight  is  in  the  true 
meridian. 

The  stadia  rod  may  be  of  the  target  variety  or  self  reading; 
somewhat  greater  accuracy  may  perhaps  be  attained  by  the 


108 


TOPOGRAPHIC    SURVEYING. 


target  rod,  buttlie  self-reading  ones  are  almost  universally  vsed. 
The  rod  is  of  pine,  about  4  inches  wide,  and  either  12  or  16  feet 
in  length ;  it  is  sometimes  stiffened  by  screwing  to 
i—  •  w  tue  back  a  longitudinal  strip  1 J  inches  square, 
\\m  while  the  ends  may  be  protected  by  a  metal  band 

U  CB      or  shoe.      There  are  numerous  designs,  but  the 
||  ^J      one  in  Fig.  50  has  been  known  to  give  good  satis- 
faction at  distances  as  great  as  2,000  feet.     The 
five-,  ten-  and  fifteen-foot  marks  are  numbered  Vt 
JTand  F"in  red,  but  the  other  foot-numbers  are 
Arabic  and  in  black.     The  bottom  and  top  of  the 
numbers  are  on  a  level  with  OJ  and  4J  tenths  so  as 
to  assist  in  readings,  and  the  triangle  marking  7^ 
tenths  is  1  tenth  on  a  side.     The  graduations  begin 
at  the  bottom,  so  that  the  rod  may  be  used  for 
leveling  as  well  as  for  stadia  work.     The  edges 
of  the  rod  are  painted  black  on  the  alternate  foot- 
marks as  shown.      The  graduations  of  the  even 
FIG.  50.        feet  are  on  the  left  side  of  the  rod,  and  those  of 
the  odd  feet  on  the  right  side. 

A  topographic  surveying  party  is  composed  of  a  transit 
man  or  observer,  a  recorder,  one  or  more  rodrnen,  and  axmen, 
if  they  are  required.  In  open  country,  where  the  topography 
is  not  very  intricate,  one  observer  can  take  sights  as  fast  as 
two  or  even  three  rodmen  can  select  points,  and  the  amount  of 
territory  covered  in  a  given  time  is  very  much  increased  by  the 
use  of  the  extra  rods  ;  in  more  difficult  territory  the  dispatch 
with  which  the  work  is  done  depends  largely  upon  the  skill  of 
the  recorder  in  keeping  his  notes  and  sketches  in  proper  shape, 
and  but  one  rodman  is  necessary.  The  work  in  the  field  con- 
sists of  running  traverse-lines  between  triangulation  stations  ; 
at  each  of  the  transit  points  along  the  traverse  the  topography 
is  taken  within  a  radius  of  500  feet  to  1000  feet  around  the 
entire  circle  in  azimuth.  The  traverses  are  so  run  that  when 
the  work  is  finished  the  entire  territory  within  the  limits  of 
the  survey  has  been  covered  by  these  circles.  Before  starting 
a  traverse-line  between  two  stations  the  elevations  of  the  sta- 
tions, the  distance  between  them,  and  the  azimuth  of  the  line 


FIELD   WORK.  109 

joining  .them  should  have  been  determined.  The  transit  is  set 
over  the  first  station,  with  the  vernier  at  the  azimuth  of  the 
line  to  the  next  triangulation  station,  and  the  telescope 
directed  to  some  point  on  that  line  ;  the  instrument  is  then 
oriented,  and  the  line  of  sight  is  brought  into  the  meridian  by 
setting  the  vernier  at  zero.  The  needle  is  allowed  to  settle 
and  the  magnetic  declination  set  off,  if  there  is  an  arrange- 
ment for  so  doing  ;  otherwise  the  reading  of  the  needle  should 
be  noted.  In  locating  the  contours  the  rod  is  held  at  every 
place  where  there  is  a  decided  change  in  the  slope  of  the 
ground  ;  in  surveying  a  small  ravine  elevations  are  taken  along 
the  valley  and  along  the  top  of  the  slope  on  each  side.  In 
work  that  is  to  be  plotted  on  a  large  scale  two  points  on  each 
building  are  located,  and  it  is  well  to  have  the  dimensions 
measured  with  a  tape.  The  rodman  should  have  a  knowledge 
of  what  it  is  desired  to  show  on  the  map,  so  that  he  need  not 
rely  upon  signals  from  the  observer  to  select  the  points  where 
observations  are  to  be  taken.  When  the  work  around  the  sta- 
tion has  been  completed,  the  rodman  selects  a  suitable  place 
for  the  next  position  of  the  transit  and  drives  a  stake  there. 
The  observer  reorients  the  transit  and  reads  the  distance  to  the 
next  stake  ;  in  determining  the  azimuth  the  edge  instead  of  the 
flat  side  of  the  rod  is  turned  toward  the  instrument.  The 
transit  is  then  set  over  the  new  station  while  the  rodman  gives 
a  backsight  on  the  last  one.  The  instrument  is  oriented  by 
directing  the  telescope  to  the  backsight,  with  the  vernier  read- 
ing the  back  azimuth  of  the  line  ;  an  easy  way  to  find  what 
the  reading  should  be  is  to  add  180°  to  azimuths  less  than  that 
amount  and  to  subtract  180°  from  those  that  are  greater.  The 
rod  reading  and  the  vertical  angle  should  be  again  observed, 
and  the  mean  of  the  two  corrected  horizontal  and  vertical  dis- 
tances is  taken  as  the  length  of  the  line  and  the  difference  in 
elevation;  the  reading  of  the  needle  may  be  used  to  detect  any 
large  errors  in  azimuth.  Below  is  given  the  manner  of  re- 
cording the  notes  on  the  left-hand  page  ;  the  right-hand  page 
is  used  for  the  sketch,  which  should  show  all  objects  located, 
and  be  as  near  to  scale  as  possible.  If  the  sketch  is  well 
made,  the  points  where  the  rod  was  held  are  numbered,  and 


110 


TOPOGRAPHIC   SURVEYING. 


the  same  numbers  appear  in  the  column  of  stations  on  the  left 
page  without  any  other  explanation.     The  traverse  is  finished 


SURVEY  o 
Instrument  at  M.    c  +  /  = 

F 

H.I.  at  M  =  491.7 
24,  1898.    Elev.  of'M  =486.6    , 

1.00.     Sept. 

Point. 

Azimuth. 

Rod 
Reading. 

Vertical 
Angle. 

Hor. 
Distance. 

Diff. 
Elev. 

Elev. 

1 
2 
3 

N 

84°  12' 
117    05 
314    42 
246    10 

907 
605 
245 
723 

-  4°  24' 
7    18 
-  0   47 
3    12 

721.8 

+  40.3 

526.9 

by  connecting  with  another  station  on  the  triangulation  system, 
which  station  should  be  occupied,  and  the  azimuth  of  the  last 
course  be  verified,  while  a  check  is  also  obtained  on  the 
elevations. 

Prob.  35.     Fill  out  the  blanks  in  the  above  field-notes  by  the 
help  of  Table  X. 


ART.  36.    OFFICE  WORK. 

The  stadia  readings  taken  between  stations  of  the  tra- 
verses are  usually  reduced  in  the  field  by  the  assistance  of 
Table  X.  The  topographer  thus  has  the  elevations  of  the 
stations  and  is  able  to  check  his  work  whenever  it  is  possible 
to  connect  with  a  station  of  known  elevation.  The  horizontal 
distances  to  minor  points  and  the  corresponding  differences  of 
level  are,  however,  often  left  to  be  filled  out  in  the  office. 
Graphical  methods  have  been  devised  for  making  these  reduc- 
tions, but  none  has  become  so  valuable  as  to  displace  the  gen- 
eral use  of  the  tables. 

The  work  of  making  the  map,  like  that  in  the  field,  is  based 
upon  the  triangulation  system,  the  stations  of  which  are  care- 
fully plotted  by  their  coordinates  as  described  in  Art.  10. 
The  traverse  lines  are  plotted  by  the  protractor,  as  by  this  way 
the  work  on  the  map  can  be  done  as  accurately  as  the  measure- 
ments were  made  in  the  field.  A  suitable  protractor  is  one  of 
wrd  hoard  12  inches  in  diameter  which  is  fastened  to  the  papor 


OFFICE   WORK.  Ill 

by  weights,  with  the  0°  and  180°  marks  on  the  meridian ; 
azimuths  are  transferred  to  any  part  of  the  map  by  means  of 
triangles  or  parallel  rulers.  If  the  work  is  carefully  done,  the 
traverse  lines  should  close  so  that  the  discrepancy  is  not  notice- 
able on  the  scale  to  which  it  is  plotted.  The  error  of  closure 
may,  with  proper  care,  be  kept  less  than  1  in  1000,  and  much 
better  results  than  this  have  been  attained. 

After  the  traverse  lines  have  been  established  the  topography 
is  plotted  by  orienting  the  protractor  over  each  station  and 
pricking  off  all  the  azimuths  of  the  readings  around  it ;  the 
protractor  is  then  removed  and  the  corresponding  distances  are 
measured  on  the  proper  scale.  The  sketch  will  show  whether 
the  point  is  merely  to  locate  contours  or  is  on  some  object  to 
be  plotted  on  the  map;  in  the  latter  case  the  house  or  whatever 
the  object  is  should  be  drawn  as  soon  as  enough  points  on  it 
have  been  established,  and  all  superfluous  marks  erased ;  if 
only  the  elevation  is  needed,  that  is  written  lightly  in  pencil. 
The  contours  cannot  be  sketched  as  fast  as  the  elevations  are 
marked,  but  this  work  should  not  be  deferred  after  enough 
heights  have  been  plotted  to  do  it  intelligently. 

What  was  stated  in  Art.  16  about  the  lettering,  title,  merid- 
ian, and  border  applies  as  well  to  topographic  drawings  and 
need  not  be  repeated.  The  execution  of  the  topographic 
signs  is  of  utmost  importance  in  determining  the  appearance 
of  the  map.  While  experienced  draughtsmen  are  able  to  dis- 
pense with  such  help,  no  student  should  attempt  to  make  the 
conventional  signs  on  a  map  without  having  before  him  a  good 
copy.  The  tendency  always  is  to  make  the  signs  much  too 
large  and  without  definite  shape.  No  amount  of  practice  will 
suffice  where  a  clear  knowledge  is  wanting  of  just  how  the 
figure  should  look. 

Prob.  36.  Draw  in  pencil  six  horizontal  lines  and  twelve 
vertical  lines  on  Fig.  43  at  equal  distances  apart.  Tlien  make 
the  same  number  of  lines  on  drawing-paper  at  distances  apart 
three  fourths  as  great.  Copy  Fig.  43  on  the  reduced  scale. 
(As  an  exercise  in  contour  drawing  Fig.  56  may  be  also  copied, 
the  scale  being  enlarged  about  one-half.) 


TOPOGRAPHIC   SURVEYING. 


AKT.  37.    THE  PLANE  TABLE. 

The  plane  table  is  a  small  drawing-board  mounted  on  a  tri- 
pod head  and  tripod  like  those  of  the  transit.  On  the  board  a 
sheet  of  paper  can  be  fastened  by  clamps.  On  the  paper  a 
heavy  ruler  may  be  placed  in  any  position.  This  ruler  is  fur- 
nished with  level  bubbles,  and  at  its  middle  has  a  standard  on 
which  is  mounted  a  telescope  provided  with  a  vertical  arc  and 
an  attached  bubble.  The  board,  which  can  be  moved  in  azi- 
muth around  the  vertical  axis  of  the  tripod  head,  corresponds 
to  the  limb  of  the  transit,  while  the  ruler  with  its  attachments 
corresponds  to  the  alidade.  The  adj  ustments  of  the  plane  table 
are  in  principle  the  same  as  those  of  the  transit.  (Art.  26). 

Although  the  plane  table  is  an  ancient  surveying  instrument, 
it  is  but  little  used  except  for  topographical  work  based  upon 
a  triangulation.  On  the  paper  are  plotted  the  stations  of  the 
triangulation,  or  as  many  as  are  contained  in  the  area  covered 
by  the  paper  on  the  scale  used.  A  common  scale  used  is  g^Vir* 
so  that  on  a  board  24  X  30  inches  in  size  an  area  of  nearly 
2  X  2£  miles  would  be  represented.  In  a  thickly  settled 
country  a  scale  of  -^-^  is  often  used. 

In  a  topographical  survey  one  of  the  first  uses  of  the  plane 
table  is  to  locate  on  the  sheet  secondary  triangulation  poCrts 

xP 


FIQ.  51. 

such  as  spires,  tall  chimneys,  or  prominent  trees.  In  Fig.  51 
this  process  is  illustrated.  A  and  B  are  two  triangulation 
stations  which  are  plotted  on  the  sheet  at  a  and  6,  and  it  is  re- 
quired to  locate  the  two  secondary  stations  G  and  D.  The 


THE    PLANE   TABLE.  113 

table  is  first  set  at  A,  the  edge  of  the  alidade  ruler  placed  upon 
the  line  ab,  the  telescope  pointed  to  B,  and  the  table  clamped 
in  position.  With  the  edge  of  the  ruler  on  a  the  telescope  is 
pointed  to  C  and  to  D,  and  indefinite  lines  drawn  in  those  direc- 
tions. The  table  is  then  set  up  at  B,  the  edge  of  the  ruler 
placed  upon  the  line  ba,  the  telescope  pointed  to  A,  and  the 
table  clamped  in  position.  With  the  edge  of  the  ruler  on  b  the 
telescope  is  pointed  to  C  and  to  D,  and  indefinite  lines  drawn  in 
those  directions.  The  intersection  of  these  with  those  pre- 
viously drawn  .at  A  gives  the  points  c  and  d,  which  are  the  loca- 
tions on  the  sheet  of  the  stations  C  and  D. 

The  operation  of  placing  the  table  so  that  each  line  on  the 
sheet  is  parallel  to  the  corresponding  line  on  the  ground  is 
called  orienting  the  table.  After  the  table  is  set  up  and  leveled 
it  must  always  be  oriented  ;  one  method  of  doing  this  is  ex- 
plained above,  and  this  will  apply  whenever  the  table  is  placed 
over  a  point  which  is  plotted  on  the  sheet  and  from  which 
other  plotted  points  can  be  seen.  The  alidade  is  often  pro- 
vided with  a  magnetic  needle  which  will  give  an  approximate 
orientation,  the  edge  of  the  ruler  being  placed  on  a  magnetic 
meridian  drawn  on  the  sheet,  and  the  table  moved  in  azimuth 
until  the  needle  points  to  JVon  the  compass  limb. 

When  the  table  is  placed'at  a  point  on  the  ground  not  plotted 
on  the  sheet,  it  is  to  be  oriented  in  general  by  the  three-point 
problem.  An  approximate  orientation  is  first  made  by  the  eye 
or  by  the  magnetic  needle.  Three  stations,  A,  B,  and  (7,  being 
visible  and  plotted  on  the  sheet  at  a,  b,  and  c,  it  is  required  to 
locate  the  point  n  corresponding  to  the  point  _ZV  over  which 
the  table  is  set.  Placing  the  alidade  ruler  on  a,  b,  and  c  in  suc- 
cession, and  sighting  on  A,  B,  and  G,  lines  are  drawn  on  the 
sheet,  and  these  intersect,  if  the  table  is  not  truly  oriented,  so 
as  to  form  a  small  triangle  of  error.  Now  the  angle  between 
the  lines  Aa  and  Bb  will  not  be  sensibly  altered  by  the  slight 
movement  necessary  to  effect  orientation  ;  hence  the  point  n 
must  lie  on  the  circumference  of  a  circle  passing  through  a,  b, 
and  the  point  of  intersection  of  these  two  lines.  Similarly,  the 
point  n  must  be  on  a  circumference  passing  through  a,  c,  and 
the  intersection  of  Aa  and  Cc.  It  is  not  practicable  to  draw 


114 


TOPOGRAPHIC   SURVEYING. 


these  circles  on  the  sheet,  but  by  imagining  them  to  be  drawn  a, 
close  estimate  of  the  point  where  they  intersect  can  be  made, 
and  n  be  marked  on  the  sheet.  Now  place  the  edge  of  the 
ruler  on  this  point  n,  and  also  on  a,  move  the  table  until  A  is 
seen  on  the  telescope  hair,  and  a  closer  orientation  is  secured. 
Then  sighting  to  B  and  (7,  and  drawing  new  lines  Bb  and  Cc,  a 


FIG.  52. 


smaller  triangle  of  error  results,  from  which  a  better  position 
of  n  is  found,  and  on  the  third  trial  the  triangle  of  error  should 
entirely  vanish,  thus  giving  both  a  correct  orientation  and  the 
proper  location  of  n  corresponding  to  N  on  the  ground. 

It  should  be  remarked  that  if  the  table  is  set  up  within  the 
large  triangle  ABC,  as  in  the  first  diagram  of  Fig.  52,  the 
point  n  falls  within  the  triangle  of  error.  In  other  cases  it 
falls  outside  the  triangle  of  error.  If  N  is  situated  on  the  cir- 
cumference of  a  circle  passing  through  A,  B,  and  (7,  the  prob 
lem  is  indeterminate,  and  another  station  D  must  be  observed  in 
connection  with  two  of  the  others.  For  a  fuller  discussion  of 
the  three-point  method  of  orientation  see  ' '  A  Treatise  on  the 
Plane  Table,"  in  Appendix  No.  13  of  the  Report  of  the  U.  S. 
Coast  and  Geodetic  Survey  for  1880. 

After  the  plane  table  is  oriented  the  topography  for  several 
hundred  feet  around  the  station  is  put  in  with  the  help  of  the 
alidade  and  stadia  rods.  The  alidade  ruler  gives  the  direction 
of  any  object,  and  the  stadia  reading  its  distance,  so  that  it  may 
be  immediately  plotted  by  a  scale  and  a  pair  of  dividers.  For 
an  inclined  stadia  reading  the  vertical  angle  is  read,  and  th 
corresponding  horizontal  and  vertical  distances  at  once  taken 
from  a  table,  the  latter  giving  the  elevation  of  the  observed 


THE   THREE-POINT  r-ROBLEM.  115 

point  above  the  table,  wliicli  is  noted  on  tlie  sheet,  so  that  the 
contours  can  be  afterward  sketched.  In  fact,  all  the  operations 
are  similar  to  those  explained  in  Art.  33,  except  that  no  notes 
are  kept.  Traverses  may  be  run  along  roads,  or  into  localities 
where  no  triangulation  points  are  visible,  by  drawing  the  lines 
successively  on  the  sheet  and  moving  the  table  from  one  station 
to  another,  orienting  it  by  a  back  sight.  Thus  the  entire  map 
is  finished  in  pencil  in  the  field.  The  theory  of  all  the  opera- 
tions is  simple,  but  the  practice  requires  some  skill  and  experi- 
ence, and  the  sheet  is  sometimes  liable  to  become  injured  by 
dust  or  rain.  Much  more  topographic  work  is  done  with  the 
transit  and  stadia  than  with  the  plane  table. 

The  three-point  problem,  above  mentioned,  also  arises  in 
secondary  triangulation  when  a  new  station  is  to  be  established 
by  means  of  angles  there  measured  between  lines  drawn  to 
three  stations,  whose  positions  are  given.  Thus  if  the  co-ordi- 
nates of  three  stations  A,  B,  and  C  are  given,  and  JV^be  the  sta- 
tion where  the  angles  ANB  and  BNG  are  measured,  then  the 
co-ordinates  of  N  can  be  computed.  Formulas  for  doing  this 
are  given  in  works  on  higher  surveying  ;  see  Merriman's 
Precise  Surveying  and  Geodesy  (New  York,  1899).1 

Prob.  37.  Given  two  stations  A  and  B,  which  are  plotted  on 
the  sheet  at  a  and  6.  It  is  required  to  set  the  plane-table  at 
two  other  points  D  and  E,  and  to  locate  d  and  e  on  the  sheet  by 
sighting  at  A,  Bt  Et  and  D. 


ART.  88.    HYDROGRAPHIC  SURVEYING. 

When  a  topographic  survey  embraces  rivers,  harbors,  or  a 
part  of  the  coast,  the  shore-lines  are  located  and  plotted  by  the 
methods  above  described.  It  is  also  generally  necessary  to 
indicate  on  the  map  the  depths  of  water  at  various  points,  the 
position  of  shoals,  rocks,  and  other  sub-surface  features,  and 
also  sometimes  to  determine  the  direction  and  velocity  of  cur- 
rents; this  part  of  the  work  cons*i*vites  hydrographic  sur- 
veying. 


116  TOPOGRAPHIC   SURVEYING. 

Soundings  in  shallow  water  are  made  by  means  of  rods  gradu- 
ated  to  feet  and  tenths.  When  the  current  is  not  rapid,  a  boat 
may  be  rowed  at  a  uniform  speed  in  a  straight  line,  which  is 
determined  by  signals  set  in  range  on  shore,  and  soundings  be 
taken  at  uniform  intervals  of  time.  The  position  of  the  boat 
both  at  the  start  and  finish  is  located  by  intersections  from 
other  signals  on  shore  or  by  means  of  observations  with  tran- 
sits. When  this  line  is  plotted  on  the  map,  it  is  divided  into 
the  same  number  of  spaces  as  there  were  time  intervals,  and  at 
each  point  of  division  the  corresponding  sounding  is  plotted. 
If  the  number  of  soundings  is  sufficient,  con  tour  curves  for  dif- 
ferent depths  below  the  water-level  may  be  drawn,  and  thus  a 
clear  picture  is  presented  of  the  bottom  surface  of  the  river  or 
harbor 

In  deep  water  where  a  rod  cannot  be  used  depths  are  obtained 
with  a  plummet  attached  to  a  line,  the  position  of  each  sound 
ing  being  located  by  angles  taken  either  on  the  boat  between 
signals  on  the  land,  or  by  observers  on  shore.  In  the  former 
case  the  sextant  is  generally  used,  two  angles  being  measured 
between  three  known  stations.  This  is  a  case  of  the  three- 
point  problem  (Art.  37).  In  plotting  the  position  from  the  two 
observed  angles  computations  are  rarely  necessary,  but  tlmie 
lines  may  be  drawn  on  tracing-cloth,  intersecting  at  a  point  and 
making  with  each  other  the  given  angles  ;  then  placing  the 
tracing  on  the  map  so  that  the  three  lines  pass  through  the 
given  stations  the  point  will  fall  in  the  proper  position  and  may 
be  pricked  through  upon  the  map. 

In  all  cases  of  sounding  a  water-gauge  should  be  erected  near 
the  shore  for  the  purpose  of  observing  the  variations  in  the 
water-level,  and  thus  referring  the  soundings  to  the  same  plane, 
either  of  high  or  of  low  water.  In  tidal  streams  or  harbors  read- 
ings of  such  a  gauge  are  necessary  at  quarter-hour  intervals. 

The  sextant  is  a  most  useful  instrument  in  all  work  done  in 
the  boat,  where  indeed  measurement  of  angles  with  a  transit 
would  be  almost  impracticable.  The  principle  of  its  use  is  that 
an  object  may  be  seen  both  by  direct  vision  and  by  reflection 
from  a  mirror.  For  instance,  in  the  first  diagram  of  Fig.  53  let 
£fand/be  two  parallel  mirrors  called  the  horizon  glass  and 


HYDROGRAPHIC  SURVEYINGS 


11? 


FIG.  63. 


118 


TOPOGRAPHIC   SURVEYING. 


the  index  glass,  the  upper  part  of  H having  an  opening  in  ft. 
Then  the  eye  at  E  can  see  a  distant  object  8,  both  by  direct 
vision  in  the  line  SHE,  and  by  the  reflected  ray  which  follows 
the  path  SIHE\  in  this  position  the  two  images  coincide  and 
the  index  arm  IA  indicates  zero  on  the  graduated  limb.  In  the 
second  diagram  the  index  arm  is  moved  to  the  position  ID  in 
order  to  measure  the  angle  SET,  between  two  signals  8  and  T ; 
in  this  position  T  is  seen  by  direct  vision  and  S  by  reflection. 
As  the  angles  of  incidence  and  of  reflection  are  equal  on  each 
mirror,  the  angle  AID  is  one  half  the  angle  SET.  The  arc  is 


FIG.  54. 


hence  graduated  so  that  half  a  degree  on  it  represents  a  whole 
degree  of  the  measured  angle  ;  thus  the  reading  at  D  gives  at 
once  the  required  angle  SET. 

In  measuring  a  horizontal  angle  the  plane  of  the  sextant 
should  be  kept  as  nearly  horizontal  as  possible.  Care  should 
be  taken  that  the  reading  of  the  vernier  is  zero  when  an  object 
is  viewed  both  by  direct  and  reflected  vision,  as  in  the  first  dia- 
gram of  Fig.  54 ;  if  this  is  not  the  case,  the  index  error  should 
be  noted  and  be  applied  as  a  correction  to  the  final  reading. 

The  direction  of  currents  may  be  noted  by  observing  with  the 
sextant  the  direction  taken  by  a  float  thrown  from  a  boat,  and 
the  velocity  of  the  current  may  be  found  by  noting  the  time 
required  for  the  float  to  pass  over  a  certain  distance.  The  de- 
termination of  velocities  at  points  below  the  surface,  and  the 
gauging  of  streams  to  ascertain  their  discharge  and  mean  veloc- 


MINE   SURVEYING.  119 

Ity,  is  properly  a  branch  of  hydraulics  rather  than  of  survey- 
ing. Concerning  these  see  Merriman's  Treatise  on  Hydraulics 
(New  York,  1916),  Chapter  10. 

Fig.  53  shows  a  part  of  a  hydrographic  map  of  the  Delaware 
River  on  a  scale  of  ^^ J^,  reproduced  from  the  chart  of  the  U. 
S.  Coast  and  Geodetic  Survey.  The  numbers  in  the  central 
part  of  the  river  show  the  depths  in  fathoms  at  mean  low- water 
spring  tides,  those  on  the  shaded  surface  show  depths  in  feet. 
The  various  lights  and  buoys  are  represented  in  proper  posi- 
tion. The  topography  of  the  shores  is  a  fine  example  of  small 
scale  work,  although  the  copy  does  not  fully  represent  the 
beauty  of  the  original  copper-plate  chart. 

Prob.  38.  Prove  that  in  Fig.  54  the  angle  AID,  moved 
over  by  the  index  arm,  is  one  half  the  observed  angle  SET. 


ART.  39.    MINE  SURVEYING. 

Mine  surveying  is  little  more  than  ordinary  surveying,  ren- 
dered difficult  by  darkness  and  mud.  The  main  object  is  to 
take  measurements  which  will  furnish  accurate  maps  of  the 
underground  workings,  so  that  the  position  of  every  point  may 
be  known  relatively  to  points  on  the  surface.  These  maps  are 
necessary,  both  ijpr  the  advantageous  development  of  the  mine 
in  driving  tunnels,  slopes,  and  gangways,  and  for  the  safety  of 
the  miners.  The  maps  of  the  anthracite  coal  regions  of  Penn- 
sylvania are  required  by  law  to  be  drawn  on  a  scale  of  100  feet 
to  1  inch,  and  to  be  kept  up  as  the  work  progresses. 

Mine  maps  show  the  main  features  of  the  surface  of  the 
ground,  such  as  streets  and  houses,  with  all  the  breakers, 
slopes,  man  way  and  air-shaft  openings.  The  underground 
workings  are  shown  in  horizontal  projection  and  proper  posi- 
tion on  the  same  sheet,  different-colored  inks  being  sometimes 
used  to  distinguish  the  different  veins.  Elevations  of  many 
points  of  the  underground  workings  are  given  in  figures,  so 
that  the  difference  of  level  between  them  and  the  surface  is 
at  once  known,  as  well  as  the  grades  of  the  gangways  and 
other  passages.  Sometimes  the  surface  contours  are  also  shown, 


120  TOPOGRAPHIC   SURVEYING. 

and  by  the  help  of  these,  and  the  elevations  of  the  underground 
points,  profiles  and  cross-sections  may  be  drawn  on  different 
vertical  planes. 

The  general  methods  of  mine  surveying  are  the  same  as 
those  of  land  and  topographical  surveying.  The  most  approved 
plan  is  to  have  on  the  surface  triangulation  stations  referred  to 
a  system  of  coordinates  (Art.  30).  At  some  mines,  however, 
coordinate  lines  are  actually  staked  out  on  the  surface.  Start 
ing  at  any  station,  a  traverse  may  be  run  down  a  slope  and 
through  a  gangway,  coming  out  perhaps  at  another  slope  or 
manway,  and  checking  on  another  triangulation  station.  This 
traverse  is  run  by  the  transit  and  a  long  steel  tape,  two  con- 
secutive stations  of  the  traverse  being  generally  nearer  to- 
gether than  the  length  of  the  tape.  Offsets  are  taken  to  the 
sides  of  the  slopes  and  gangways,  and  short  lines  are  run  up 
the  breasts  and  openings.  Thus  all  the  data  are  obtained  for 
computing  the  traverse  and  constructing  the  map.  Elevations 
are  determined  by  taking  vertical  angles,  although  when  con- 
venient the  level  and  rod  is  sometimes  used. 

The  stations  of  the  underground  traverse  are  placed  in  the 
roof  on  wooden  plugs  driven  into  holes  drilled  for  that  pur- 
pose. On  these  are  hung  the  plummet  lamps  to  which  back- 
sights and  foresights  are  taken.  To  set  up  the  transit  at  a 
station  a  point  on  the  floor  directly  beneath  the  one  in  the  roof 
is  determined  by  the  plumb-bob.  A  transit  for  mine  surveys 
should  have  a  shifting  plate  and  adjustable  tripod  legs,  while 
a  universal  joint  is  also  often  a  great  convenience.  To  illumine 
the  cross-wires  the  transitman  holds  his  copper  lamp  at  arrn's- 
length  so  that  the  light  may  shine  into  the  objective  end  of  the 
telescope  ;  the  same  lamp  enables  him  to  read  the  vernier  and 
the  magnetic  needle.  The  readings  of  the  magnetic  needle, 
which  serve  as  checks  on  the  horizontal  angles,  must  be  taken 
both  backward  and  forward  at  each  station,  as  marked  local 
attractions  occur  in  mines.  Much  time  is  often  wasted  in 
reading  the  needle  ;  instead  it  would  be  better  to  check  the  azi- 
muth by  taking  another  angle.  The  linear  measurements  are 
made  when  the  tape  is  tightly  stretched  by  two  men,  offsets 


MINE   SURVEYING. 


FIG.  55. 


122  TOPOGRAPHIC  SURVEYING. 

being  taken  to  the  corners  of  pillars  and  tlie  sides  of  the  gang- 
ways. A  mine  survey  corps  usually  consists  of  four  or  five 
men,  a  transitman,  two  chainmen,  and  one  or  two  men  for  off- 
sets and  lights. 

The  form  of  field-notes  may  be  the  same  as  that  given  in  Art. 
15,  but  instead  of  measuring  the  interior  angles  it  is  best  to 
carry  on  the  azimuths  as  explained  in  Art.  19.  Some  prefer 
to  reverse  the  telescopes  and  measure  the  deflection  angle  to  the 
right  or  left,  but  this  is  inferior  in  accuracy  and  convenience 
to  the  method  of  azimuths.  The  form  of  notes  is  subject  to  so 
great  variations  in  different  localities,  that  it  seems  scarcely 
wise  to  attempt  to  give  one  of  them  here. 

The  computation  of  the  coordinates  of  the  stations  of  the 
traverse  is  next  made.  Lines  being  drawn  on  the  paper  500 
feet  apart  both  vertically  and  horizontally,  the  stations  are 
plotted  in  their  proper  positions.  The  offsets  are  then  laid  off 
and  the  sides  of  the  slopes,  gangways,  air-passages,  and  breasts 
are  drawn.  The  underground  traverse-lines  are  usually  plotted 
in  red,  and  each  station  designated  by  its  letter  or  number. 
The  elevations  are  noted  in  figures  at  such  stations  where 
they  may  be  likely  to  be  needed.  If  surface  features  are 
to  be  also  given,  they  are  plotted  from  the  notes  of  an  outside 
survey. 

Fig.  55  shows  a  part  of  a  map  of  an  anthracite  coal  mine,  re- 
duced from  the  original  scale  of  100  feet  to  1  inch  to  about 
half  that  scale.  It  shows  the  buildings  around  a  slope  entrance, 
and  the  slope  with  a  few  gangways  and  breasts.  The  fine 
broken  lines  are  the  traverses  of  the  survey  and  each  station 
has  its  number  ;  a  traverse  is  seen  to  start  at  A  near  the  pump 
house,  run  down  the  slope  to  station  4,  and  then  turn  to  the 
west  along  the  upper  lift  gangway.  The  long  pillars  seen  in 
each  gangway  separate  it  from  the  air  way.  In  every  fifth 
breast  is  written  the  number  by  which  it  is  known. 

Extended  surface  surveys  in  the  mining  regions  come  under 
the  head  of  topography  taken  with  especial  reference  to  geo- 
logic features.  Fig.  56  shows  a  small  area  near  Carbondale, 
Pa.,  taken  from  Mine  Sheet  No.  XXI  of  Part  IV  of  the  Atlas  of 


MIKE  SURVEYING. 


123 


the  Northern  Anthracite  Coal  Field,  issued  by  the  Second  Geo- 
logical Survey  of  Pennsylvania.  The  scale  is  1  inch  to  800  feet 
and  the  contour  interval  is  10  feet,  the  elevations  being  given 
with  reference  to  tide  water.  The  coordinate  lines,  drawn  at 
intervals  of  2000  feet,  give  distances  north  and  east  from  a 


FIG.  56. 


monument  in  the  yard  of  the  court-house  at  Wilkes  Barre. 
Bore-holes,  dips  of  strata,  and  outcrops  of  the  formations  are 
shown,  as  also  property  lines,  and  names  of  owners  or  lessees. 
The  colors  on  the  original  map  are  not  reproduced  in  the 
copy. 


124  TOPOGRAPHIC   SURVEYING. 

Prob.  29.     By  surveys  and  computations   the  following  data 
were  obtained   concerning  four  points  in  a  certain   gangway 
driven  around  one  end  of  a  vein  in 
a  coal  basin: 

Station.  Latitude.  Longitude. 

A  +2604.25  +2428.10 

B  +  2597.18  +  2010.43 

N  +  3345.65  +  2904.18 

Also,  elevation  of  A  =  783.84,  ele> 
vation  of  N  =  807.90,  azimuth  of 
MN  =  92°  17'  (S  87°  43'  E).  It  is 
desired  to  drive  a  tunnel  from  A  to 
N,  and  for  this  purpose  the  follow- 
ing quantities  are  required  to  be 
found  :  (1)  Length  of  line  AN,  (2)  azimuth  of  AN,  (3)  the 
horizontal  angle  BAN,  (4)  the  horizontal  angle  MNA,  (5)  the 
grade  of  the  line  AN. 

ART.  40.    THE  TRUE  MERIDIAN. 

A  true  meridian  is  established  by  actually  staking  out  a  line 
running  due  north  and  south,  or  by  determining  the  true  azi- 
muth of  a  given  line.  The  latter  method  is  preferable  in  tow?i 
and  city  work.  From  the  azimuth  found  for  the  one  line  th'3 
azimuths  of  all  other  important  lines  are  obtained  by  travers- 
ing or  by  triangulation.  A  meridian  actually  staked  out  is  of 
no  value  except  for  determining  the  azimuths  of  lines.  Three 
methods  of  determining  the  true  meridian  will  be  here  ex- 
plained. 

By  Polaris  and  Mizar. — The  pole-star  Polaris  revolves 
around  the  pole  in  a  small  circle,  and  crosses  the  meridian,  or 
culminates,  twice  each  day.  Mizar,  the  middle  one  of  the  three 
stars  in  the  tail  of  the  Great  Bear  or  handle  of  the  Great 
Dipper,  revolves  around  the  pole  in  a  large  circle  and  culmi- 
nates a  few  minutes  earlier  than  Polaris.  In  1895  Polaris  cul- 
minates about  50  seconds  after  it  and  Mizar  are  in  the  aarne 
vertical  circle,  in  1900  about  2£  minutes  after,  and  in  1905 
about  4£  minutes  after,  the  annual  increase  being  21  seconds. 
To  obtain  the  true  meridian  set  up  a  transit  about  a  quarter  of 
an  hour  before  the  two  stars  are  in  the  same  vertical ;  the 


THE  TRUE   MERIDIAK.  125 

transit  must  be  in  good  adjustment,  particularly  in  respect  to 
collimation  and  horizontal  axis  of  the  telescope.  Sight  alter- 
nately upon  Polaris  and  Mizar,  and  note  by  a  watch  the  time 
when  they  are  upon  the  same  vertical.  Then,  after  the  expi- 
ration of  the  interval  above  mentioned,  turn  the  vertical  hair 
upon  Polaris,  and  the  line  of  sight  coincides  with  the  true 
meridian.  The  error  of  this  method  will  probably  be  greater 
than  one  minute  of  angle,  as  the  work  must  be  done  at  night. 

By  Polaris.  — The  time  of  culmination  of  Polaris  may  be  as- 
certained from  Table  V,  and  the  vertical  hair  of  a  transit  be 
set  upon  it  at  that  instant.  But  a  more  accurate  method  is  to 
observe  Polaris  at  its  east  or  west  elongation,  following  it  with 
Ihe  vertical  hair  until  its  motion  in  azimuth  ceases.  The  ap- 
proximate time  of  elongation  may  be  found  from  Table  V,  and 
!,he  astronomical  azimuth  of  Polaris  at  elongation  is  found  from 
Table  VI.  Thus  the  azimuth  of  the  line  of  sight  is  known  ;  if 
to  point  be  marked  beneath  the  plumb-bob  and  another  several 
hundred  feet  away  in  the  line  of  sight,  a  line  is  determined 
whose  azimuth  is  known.  By  repeating  the  operation  on  sev- 
eral days  a  mean  result  can  be  obtained  which  can  be  depended 
upon  with  an  error  not  exceeding  one  minute  of  angle.  This 
work  need  not  be  done  at  night,  as  Polaris  can  often  be  seen 
by  a  telescope  of  moderate  power  in  the  daytime. 

By  the  Sun. — With  a  transit  having  a  solar  attachment  the 
true  meridian  can  be  found  by  observing  the  sun  at  any  time 
except  between  11  A.M.  and  1  P.M.  Such  an  attachment  can  be 
placed  upon  any  transit  at  a  cost  of  about  fifty  dollars.  Accom- 
panying it  is  a  pamphlet  giving  full  directions  for  use  and 
adjustment,  together  with  tables  of  the  declination  of  the  sun 
for  Greenwich  noon  on  each  day  of  the  year.  Both  the  transit 
and  the  solar  attachment  should  be  in  correct  adjustment  in  or- 
der to  do  good  work  in  determining  the  true  meridian. 

In  order  to  explain  the  theory  of  the  solar  attachment  let  the 
upper  part  of  Fig.  58  be  a  section  of  the  celestial  sphere  in  the 
plane  of  the  true  meridian,  JV  and  S  being  the  north  and  south 
points  of  the  horizon,  P  the  pole,  Z  the  zenith,  Q  the  celes- 
tial equator,  and  0  the  place  of  the  sun  at  noon.  Let  A  be 
the  point  where  the  instrument  is  set,  which  may  be  regarded 


126  TOPOGRAPHIC    SURVEYING. 

as  the  center  of  the  celestial  sphere.  Then  the  angle  PAN  or 
its  equal  QAZ  is  the  latitude  of  the  place  of  observation.  The 
angle  QAO  is  the  declination  of  the 
sun,  which  is  positive  when  the 
sun  is  north  of  the  equator  from 
March  21  to  September  21,  and  neg- 
ative when  the  sun  is  south  of  the 
N  equator  from  September  21  to 
March  21.  The  lower  part  of  Fig. 
58  is  a  plan,  A  being  the  place  of 
the  instrument,  WS  the  true  me- 
Fia.  58.  ridian  through  A,  W  and  E  the 

west  and  east  directions,  AO  the  direction  of  the  sun  about 
10  o'clock  in  the  morning,  and  AL  a  line  whose  azimuth  is 
required  to  be  found. 

Let  ab  represent  the  telescope  of  the  transit,  placed  in  the 
meridian  and  elevated  so  as  to  point  to  the  celestial  equator ; 
this  will  be  the  case  when  the  angle  of  elevation  SAQ  is  equal 
to  the  co-latitude,  or  when  SAQ  =  90°  —  QAZ.  Let  cd  be 
the  telescope  of  the  solar  attachment  pointing  toward  the  sun; 
then  the  vertical  angle  between  ab  and  cd  is  equal  to  the  dec- 
lination of  the  sun  QAO.  In  this  position  the  solar  attach- 
ment is  like  an  equatorial  telescope,  its  axis  pointing  to  the 
pole  P,  and  as  the  sun  moves  the  telescope  cd  will  follow  it 
along  the  celestial  sphere  until  the  change  in  declination  be- 
comes appreciable. 

Before  beginning  work  a  list  of  hourly  declination  settings 
is  to  be  prepared  by  help  of  the  table  of  declinations  which  is 
furnished  by  the  maker  of  the  instrument.  This  table  also 
gives  for  each  hour  the  effect  of  refraction,  this  refraction  al- 
ways increasing  the  altitude  of  the  sun.  For  example,  let  it 
be  required  to  find  the  declination  settings  for  the  afternoon  of 
September  19,  1895,  for  any  place  where  eastern  standard  time 
is  used.  The  table  gives  +  1°  28'  54"  as  the  declination  of  the 
Bun  at  Greenwich  noon  for  that  day,  and  58"  as  the  hourly  de- 
crease of  declination.  The  declination  at  7  A.M.  of  eastern 
standard  time  is  then  +  1°  28'  54",  and  that  at  5  P.M.  is 
+ 1°  28'  54"  —  10  X  58"  =  -J- 1°  21'  14".  Thus  the  declination 


THE   TRUE   MERIDIAN. 


127 


for  each  hour  is  found  and  given  in  the  second  column.  In  the 
third  column  is  placed  the  refraction  correction  as  given  in  the 
table,  and  the  fourth  column  gives  the  final  declination  settings 


Hour. 

Declination. 

Refraction 
Correction. 

Declination 
Settings. 

Remarks. 

1  P.M. 

4-  1°  25'  06" 

4  0'  48" 

+  1°  25'  54" 

For  Eastern 

2  P.M. 

4-  1     24    08 

+  0    54 

41    24   52 

Standard  Time, 

3  P.M. 

+  1    23    10 

4  1    05 

+  1     24    15 

September  19, 

4  P  M. 

4-  1     2:2    12 

4  1    32 

4-1    23    44 

1895. 

5PM. 

4  1     21    14 

42    51 

41    23    05 

which  are  the  apparent  declinations  for  the  respective  hours. 
The  refraction  correction  is  always  additive,  and  hence  if  the 
declination  is  south  or  negative  its  numerical  value  is  decreased, 


Hour. 

Declination. 

Refraction 
Correction. 

Declination 
Settings. 

Remarks. 

8A.M. 
9  A.M. 
10A.M. 
1.1  A.M. 

—  22°  23'  43" 
—  22    24    02 
—  22    24    21 
—  22   24    40 

4  6'  31" 
4-  2    59 
4  2    11 
4  1    54 

—  22°  17'  12" 
—  22    21    03 
—  22    22    10 
—  22    22    46 

For  Eastern 
Standard  Time, 
December  5, 
1895. 

as  the  example  for  December  5,  1895,  shows  ;  on  that  day  the 
table  gives  the  declination  at  Greenwich  noon  as  22°  23'  24" 
south  and  the  hourly  change  as  19  seconds. 

After  this  list  is  made  out  the  observer  sets  up  the  transit 
over  the  point  A  in  order  to  find  the  true  azimuth  of  a  line  AL 
(Fig.  58).  The  telescope  is  leveled  by  the  attached  bubble  and 
pointed  approximately  toward  the  south.  The  declination  set- 
ting for  the  hour  is  next  laid  off  on  the  vertical  arc,  depress- 
ing the  object  glass  if  the  declination  is  positive  and  elevating 
it  if  the  declination  is  negative.  The  telescope  of  the  solar  is 
then  leveled  by  means  of  its  own  bubble,  and  thus  the  angle 
between  the  two  telescopes  is  the  same  as  the  apparent  decli- 
nation of  the  sun  QAO.  Both  telescopes  are  then  elevated  until 
the  vertical  arc  reads  an  angle  equal  to  the  co-latitude  of  the 
place,  or  8 A  Q.  The  solar  attachment  is  next  turned  on  its  axis, 
and  the  limb  of  the  transit  upon  its  axis,  until  the  sun  is  seen 
inscribed  in  the  square  formed  by  the  four  extreme  cross-hairs 


128 


TOPOGRAPHIC   SURVEYING. 


in  the  focus  of  the  solar  telescope.  When  this  is  the  case,  the 
transit  telescope  is  in  the  plane  of  the  meridian,  and  if  desired 
a  point  may  be  set  out  in  the  line  AS  to  mark  that  meridian. 

It  will  be  better,  however,  to  read  both  verniers  on  the  hori- 
zontal circle,  then  turn  the  alidade  around  to  L  and  read  both 


Time. 

Reading  on 
Meridian. 

A.     B. 

Reading  on  L. 
A.    B. 

Angle  SAL, 

Remarks. 

9:15  A.M. 
9:3C 
9:45 
3:15  P.M. 
3:30 
4:00 

20°  19' 
8000 
140  59 
200  01 
•260  13 
320  06 

00" 
15 
30 
60 
45 
00 

30" 
15 
15 
45 
30 
00 

182°27' 
242  08 
303  08 
2  09 
62  21 
122  14 

30" 
30 
45 
45 
15 
45 

30" 
00 
15 
30 
30 
60 

162°  08'  15" 
162    09    00 
162    09    08 
162    07    45 
162    08    45 
162    08    53 

Oct.  28,  1895. 
R.  Doe, 
Observer. 
Mean  = 
162°  08'  38" 
Azimuth  AL 
=  17°  51'  22" 

verniers  again.  The  angle  SAL,  which  is  the  azimuth  of  L, 
has  thus  been  measured.  Repeating  again  the  operation  with 
the  solar  another  value  of  SAL  is  determined,  and  by  making 
several  measures,  both  in  the  morning  and  afternoon,  the  mean 
result  can  be  relied  upon  with  a  probable  error  of  about  one 
minute  if  the  observer  be  skilled  in  such  work.  The  above 
form  indicates  a  method  of  keeping  the  field-notes. 

By  an  Altitude  of  the  Sun. — The  altitude  of  the  sun  may  b« 
taken  with  a  common  transit,  and  this,  together  with  the 
declination  of  the  sun  and  the  latitude  of  the  place,  gives  the 
means  of  computing  the  azimuth  of  the  sun  at  the  moment  of 
observation.  This  method  is  explained  in  full  on  page  243. 

ART.  41.    ISOGONIC  MAP  OF  UNITED  STATES. 

An  Isogonic  Line  on  a  map  is  a  curve  passing  through  all 
places  where  the  magnetic  needle  has  the  same  declination. 
The  chart  on  the  next  page  shows  these  lines  for  the  United 
States  on  January  1,  1915.  At  all  places  on  the  line  marked 
0°  the  magnetic  needle  then  had  no  declination;  that  is,  its 
north  end  pointed  to  the  true  north.  East  of  the  0°  the  north 
end  of  the  needle  pointed  west  of  the  true  north  and  west  of 
the  0°  line  it  pointed  west  of  the  true  north.  Thus  at  Boston, 
Mass.,  the  declination  in  1915  was  about  14°  W,  and  at  Helena, 
Mont.,  it  was  about  21°  E. 


ISOGOSTIC   MAP   OF   U.  S. 


1280 


PLATE  I.    ISOGONIC  MAP  OF  U.  S.  FOR  1915. 


1285        TOPOGRAPHIC  SURVEYING. 

These  isognoraic  lines  are  constantly  shifting;  the  0°  line  is 
moving  westward  at  a  rate  between  1'  and  2'  per  year.  On 
the  chart  two  parallel  lines  are  seen  extending  through  the 
middle  west;  at  all  places  on  that  double  line  there  was  no 
yearly  change  in  declination  in  1915;  at  all  places  east  of  that 
double  line  the  west  declination  was  increasing;  at  all  places 
westward  the  eastern  declination'was  decreasing.  Thus,  near 
Denver,  Colo.,  the  east  declination  in  1915  was  decreasing  at 
the  rate  of  about  3'  per  year. 

A  rough  estimate  of  the  magnetic  declination  for  any  place 
for  any  year  between  1910  and  1920  can  be  made  by  the  help 
of  this  chart.  Thus,  for  Washington,  D.C.,  the  chart  gives 
the  declination  in  1915  as  6°  W  and  the  annual  change  as 
4'.  4  W;  hence  the  change  in  five  years  was  22'  W  or  about 
0°.4  W,  and  accordingly  the  declination  in  1910  was  ap- 
proximately 5°. 6  "W  and  that  in  1920  will  be  approximately 
6°.4  W.  An  estimate  of  this  kind  cannot  be  relied  upon 
within  0°.3. 


TABLE  I. 


NATURAL  SINES  AND   COSINES 

TO 

FIVE  DECIMAL  PLACES. 


130 


TABLE   I.      SINES   AND   COSINES. 


0° 

1° 

2° 

3° 

4° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

0 

.00000 

One. 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

.99756 

60 

1 

.00029 

One. 

.01774 

.99984 

.03519 

.99938 

.05263 

.99861 

.07005 

.99754 

59 

2 

.00058 

One. 

.01803 

.99984 

.03548 

.99937 

.05292 

.99860 

.07034 

.99752 

58 

3 

.00087 

One. 

.01832 

.99983 

.03577 

.99936 

.05321 

.99858 

.07063 

.99750 

57 

4 

.00116 

One. 

.01862 

.99983 

.03606 

.99935 

.05350 

.99857 

.07092 

.99748 

56 

5 

.00145 

One. 

.01891 

.99982 

.03635 

.99934 

.053/9 

.99855 

.07121 

.99746 

55 

6 

.00175 

One. 

.01920 

.99982 

.03664 

.99933 

.05408 

.99854 

.07150 

.99744 

54 

7 

.00204 

One. 

.01949 

.99981 

.03693 

.99932 

.05437 

.99852 

.07179 

.99742 

53 

8 

.00233 

One. 

.01978 

.99980 

.03723 

.99931 

.05466 

.99851 

.07208 

.99740 

52 

9 

.00262  One. 

.02007 

.99980 

.03752 

.99930 

.05495 

.99849 

.07237 

.99738 

51 

±0 

.002911  One. 

.02036 

.99979 

.03781 

.99929 

.05524 

.99847 

.07266 

.99736 

50 

11 

.00320  .99999 

.02065 

.99979 

.03810 

.99927 

.05553 

.99846 

.07295 

.99734 

49 

12 

.00349  .99999 

.02094 

.99978 

.03839 

.99926 

.05582 

.998441 

.07324 

.99731 

48 

13 

.00378  .99999 

.02123 

.99977 

.03868 

.99925 

.05611 

.99842; 

.07353 

.99729 

47 

14 

.00407  .99999 

.02152 

.99977 

.03897 

.99924 

.05640 

.99841 

.07382 

.99727 

46 

15 

.00436  .99999 

.02181 

.99976 

.03926 

.99923 

.05669 

.99839 

.07411 

.99725 

45 

16 

.00465  .99999 

.02211 

.99976 

.03955 

.99922 

.05698 

.99838 

.07440 

.99723 

44 

17 

.00495  .99999 

.02240 

.99975 

.03984 

.99921 

.05727 

.99836 

.07469 

.99721 

43 

18 

.00524  .99999 

.02269 

.99974 

.04013 

.99919 

.05756 

.99834 

.07498 

.99719 

42 

19 

.00553  .99998 

.02298 

.99974 

.04042 

.99918 

.05785 

.99833 

.07527 

.99716 

41 

20 

.  00582  j.  99998 

.02327 

.99973 

.04071 

.99917 

.05814 

.99831 

.07556 

.99714 

40 

21 

.00611  .99998 

.02356 

.99972 

.04100 

.99916 

.05S44 

.99829 

.07585 

.99712 

39 

22 

.  00640  !.  99998 

.02385 

.99972 

.04129 

.99915 

.05873 

.99827 

.07614 

.99710 

38 

23 

.  00669  :.  99998 

.02414 

.99971 

.04159 

.99913 

.05902 

.99826 

.('7643 

.99708 

37 

24 

.00693  '.99998 

.02443 

.99970 

.04188 

.99912 

.05931 

.99824 

.07672 

.99705 

36 

25 

.00727 

.99997 

.02472 

.99969 

.04217 

.99911 

.05960 

.99822 

.07701 

.99703 

35 

26 

.00756 

.99997 

.02501 

.99969 

.04246 

.99910 

.05989 

.99821 

.07730 

.99701 

34 

27 

.00785 

.99997 

.02530 

.99968 

.04275 

.99909 

.06018 

.99819 

.07759 

.99699 

33 

28 

.00814 

.99997 

.02560 

.99967 

.04304 

.99907 

.06047 

.99817 

.07788 

.99696;  32 

29 

.00844 

.99996 

.02589 

.99966 

.04333 

.99906 

.06076 

.99815 

.07817 

.99694  31 

30 

.00873 

.99996 

.02618 

.99966 

.04362 

.99905 

.06105 

.99813 

.07846 

.99692 

30 

31 

.00902 

.99996 

.02647 

.99965 

.04391 

.99904 

.06134 

.99812 

.07875 

.99689 

29 

32 

.00931 

.99996 

.02676 

.99964 

.04420 

.99902 

.06163 

.99810 

.07904 

.99687 

28 

33 

.00960 

.99995 

.02705 

.99963 

.04449 

.99901 

.06192 

.99808 

.07933 

.99685 

27 

34 

.00989  '.99995 

.02734 

.99963 

.04478 

.99900 

.06221 

.99806 

.07962 

.99683 

26 

35 

.01018!.  99995 

.02763 

.99962 

.04507 

.99898 

.06250 

.99804 

.07991 

.99680 

25 

36 

.01047  .99995 

.02792 

.99961 

.04536 

.99897 

.06279 

.99803 

.08020 

.996781  24 

37 

.010761.99994 

.02821 

.99960 

.04565 

.99896 

.06308 

.99801 

.08049 

.99676 

23 

38 

.01  105  1.  99994 

.02850 

.99959 

.04594 

.99894 

.06337 

.99799 

.08078 

.99673 

22 

39 

.01134  .99994 

.02879 

.99959 

.04623 

.99893 

.06366 

.99797 

.08107 

.99671 

21 

40 

.01164J.  99993 

.02908 

.99958 

.04653 

.99892 

.06395 

.99795 

.08136 

.99668 

20 

41 

.01193 

.99993 

.02938 

.99957 

.04682 

.99890 

.06424 

.99793 

.08165 

.99666 

19 

42 

.01222 

.99993 

.02967 

.99956 

.04711 

.99889 

.06453 

.99792 

.08194 

.99664 

18 

43 

.01251 

.99992 

.02996 

.99955 

.04740 

.99888 

.06482 

.99790 

.08223 

.99661 

17 

44 

.01280 

.99992 

.03025 

.99954 

.04769 

.99886 

.06511 

.99788 

.08252 

.99659 

16 

45 

.01309 

.99991 

.03054 

.99953 

.04798 

.96885 

.06540 

.99786 

.08281 

.99657 

15 

46 

.01338 

.99991 

.03083 

.99952 

.04827 

.99883 

.06569 

.99784 

.08310 

.99654 

14 

47 

.01367 

.99991 

.03112 

.99952 

.04856 

.99882 

.06598 

.99782 

.08339 

.99652 

13 

48 

.01396 

.99990 

.03141 

.99951 

.04885 

.99881 

.06627 

.99780 

.08368 

.99649 

12 

49 

.01425 

.09990 

.03170 

.99950 

.04914 

.99879 

.06656 

.99778 

.08397 

.99647 

11 

50 

.01454 

.99989 

.03199 

.99949 

.04943 

.99878 

.06685 

.99776  j 

.08426 

.99644 

10 

51 

.01483 

.99989 

.03228 

.99948 

.04972 

.99876 

.06714 

.99774 

.08455 

.99642 

9 

52 

.01513  .99989 

.03257 

.99947 

.05001 

.99875 

.06743 

.99772 

.08484 

.99639 

8 

53 

.015421.99988 

.03286 

.99946 

.05030 

.99873 

.06773 

.99770 

.08513 

.99637 

7 

54 

.01571  .99988 

.03316 

.99945 

.05059 

.99872 

.06802 

.99768; 

.08542 

.99635 

6 

55 

.01600  .99987 

.03345 

.99944 

.05088 

.99870 

.06831 

.99766 

.08571 

.99632 

5 

56 

.016291.99987 

.03374 

.99943 

.05117 

.99869 

.06860  .99764 

.08600 

.99630 

4 

57 

.01658!.  99986 

.03403 

.99942 

.05146 

.99867 

.06889  .99762 

.08629 

.99627 

3 

58 

.01687 

.99986 

.03432 

.99941 

.05175 

.99866 

.06918 

.99760 

.08658 

.99625 

2 

59 

.01716 

.99985 

.03461 

.99940 

.05205 

.99864 

.06947 

.99758' 

.08687 

.99622 

1 

60 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

.  99756  j 

.08716 

.99619 

J) 

/ 

Cosin  |  Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

t 

89° 

88° 

87° 

86° 

85' 

TABLE   T.      SINES   AND   COSINES. 


131 


5° 

6° 

70 

8« 

9° 

I 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  1  Cosin 

/ 

0 

.08716 

.99619 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15643  .98769  :  60 

1 

.08745 

.99617 

.10482 

.99449 

.12216 

.99251 

.13946 

.99023 

.  15672  !.  98764:  59 

2 

.08774 

.99614 

.10511 

.99446 

.12245 

.99248 

.13975 

.99019 

.15701s.  987601  58 

3 

.08803 

.99612 

.10540 

.99443 

.12274 

.99244 

.14004 

.99015 

.  15730  !.  98755!  57 

4 

.08831 

.99609 

.10569 

.99440 

.12302 

.99240 

.14033 

.99011 

.15758  .98751!  56 

5 

.08860 

.99607 

.10597 

.99437 

.12331 

.99237 

.14061 

.99006 

.15787  .987461  55 

6 

.08889 

.99604 

.10626 

.99434 

.12360 

.99233 

.14090 

.99002 

.15816 

.98741  54 

7 

.08918 

.99602 

.10655 

.99431 

.12389 

.99230 

.14119 

.98998 

.15845 

.98737  53 

8 

.08947 

.99599 

.10684 

.99428 

.12418 

.99226 

.14148 

.98994 

.15873  .98732  52 

9 

.08976 

.99596 

.10713 

.99424 

.12447 

.99222 

.14177 

.98990 

.15902 

.  98728  '  51 

10 

.09005 

.99594 

.10742 

.99421 

.12476 

.99219 

.14205 

.98986 

.15931 

.98723 

50 

11 

.09034 

.99591 

.10771 

.99418 

.12504 

.99215 

.14234 

.98982 

.15959 

.98718 

49 

12 

.09063 

.99588 

.10800 

.99415 

.12533 

.99211 

.14263 

.98978 

.15988 

.98714  48 

13 

.09092 

.99586 

.10829 

.99412 

.12562 

.99208 

.14292 

.98973 

.16017 

.98709  47 

14 

.09121 

.99583 

.10858 

.99409 

.12591 

.99204 

.14320 

.98969 

.16046 

.98704  46 

15 

.09150 

.99580 

.10887 

.99406 

.12620 

.99200 

.14349 

.98965 

.16074 

.98700-  45 

16 

.09179 

.99578 

.10916 

.99402 

.12649 

.99197 

.14378 

.98961 

.16103 

.98695 

44 

17 

.09208 

.99575 

.10945 

.99399 

.12678 

.99193 

.14407 

.98957 

.16132 

.98690 

43 

18 

.09237 

.99572 

.10973 

.99396 

.12706 

.99189 

.14436 

.98953 

.16160 

.98686 

42 

19 

.09266 

.99570 

.11002 

.99393' 

.12735 

.99186 

.14464 

.98948 

.16189 

.98681 

41 

20 

.09295 

.99567 

.11031 

.99390 

.12764 

.99182 

.14493 

.98944 

.16218 

.98676 

40 

21 

.09324 

.99564 

.11060 

.99386 

.12793 

.99178 

.14522 

.98940 

.16246 

.98671 

39 

22 

.09353 

.99562 

.11089 

.99383 

.12822 

.99175 

.14551 

.98936 

.16275 

.98667 

38 

23 

.09382 

.99559 

.11118 

.99380 

.12851 

.99171 

.14580 

.98931 

.16304 

.98662 

37 

24 

.09411 

.99556 

.11147 

.99377 

.12880 

.99167 

.14608 

.98927 

.16333 

.98657 

36 

25 

.09440 

.99553 

.11176 

.99374 

.12908 

.99163 

.14637 

.98923 

.16361 

.98652 

35 

26 

.09469 

.99551 

.11205 

.99370 

.12937 

.99160 

.14666 

.98919 

.16390 

.98648 

34 

27 

.09498 

.99548 

.11234 

.99367 

.12966 

.99156 

.14695 

.98914 

.16419 

.98643 

33 

28 

.09527 

.99545 

.11263 

.99364 

.12995 

.99152 

.14723 

.98910 

.16447 

.98638 

32 

29 

.09556 

.99542 

.11291 

.99360 

.13024 

.99148 

.14752 

.98906 

.16476 

.98633 

31 

30 

.09585 

.99540 

.11320 

.99357 

.13053 

.99144 

.14781 

.98902 

.16505 

.98629 

30 

81 

.09614 

.99537 

.11349 

.99354 

.13081 

.99141 

.14810 

.98897 

.16533 

.98624 

29 

32 

.09642 

.99534 

.11378 

.99351 

.13110 

.99137 

.14838 

.98893 

.16562 

.98619 

28 

33 

.09671 

.99531 

.11407 

.99347 

.13139 

.99133 

.14867 

.98889 

.16591 

.98614 

27 

34 

.09700 

.99528 

.11436 

.99344 

.13168 

.99129 

.14896 

.98884 

.16620 

.98609  26 

35 

.09729 

.99526 

.11465 

.99341 

.13197 

.99125 

.14925 

.98880 

.16648 

.98604  25 

36 

.09758 

.99523 

.11494 

.99337 

.13226 

.99122 

.14954 

.98876 

.16677 

.98600  24 

37 

.09787 

.99520 

.11523 

.99334 

.13254 

.99118 

.14982 

.98871 

.16706 

.98595  23 

38 

.09816 

.99517 

.11552 

.99331 

.13283 

.99114 

.15011 

.98867 

.16734 

.98590  22 

39 

.09845 

.99514 

.11580 

.99327 

.13312 

.99110 

.15040 

.98863 

.16763 

.98585 

21 

40 

.09874 

.99511 

.11609 

.99324 

.13341 

.99106 

.15069 

.98858 

.16792 

.98580 

20 

41 

.09903 

.99508 

.11638 

.99320 

.13370 

.99102 

.15097 

.98854 

.16820 

.98575 

19 

42 

.09932 

.99506 

.11667 

.99317 

.13399 

.99098 

.15126 

.98849 

.16849 

.98570 

18 

43 

.09961 

.99503 

.11696 

.99314 

.13427 

.99094 

.15155 

.98845 

.16878 

.98565 

17 

44 

.09990 

.99500 

.11725 

.99310 

.13456 

.99091 

.15184 

.98841 

.16906 

.98561 

16 

45 

.10019 

.99497 

.11754 

•99307 

.13485 

.99087 

.15212 

.98836 

.16935 

.98556 

15 

46 

.10048 

.99494 

.11783 

.99303 

.13514 

.99083 

.15241 

.98832 

.16964 

.98551 

14 

47 

.10077 

.99491 

.11812 

.99300 

.13543 

.99079 

.15270 

.98827 

.16992 

.98546 

13 

48 

.10106 

.99488 

.11840 

.99297 

.13572 

.99075 

.15299 

.98823 

.17021 

.98541 

12 

49 

.10135 

.99485 

.11869 

.99293 

.13600 

.99071 

.15327 

.98818 

.17050 

.98536 

11 

60 

.101S4 

.99482 

.11898 

.99290 

.13629 

.99067 

.15356 

.98814 

.17078 

.98531 

10 

51 

.10192 

.99479 

.11927 

.99286 

.13658 

.99063 

.15385 

.98809 

.17107 

.98526 

9 

52 

.10221 

.99476 

.11956 

.99283 

.13687 

.99059 

.15414 

.98805 

.17136 

.98521 

8 

53 

.10250 

.99473 

.11985 

.99279 

.13716 

.99055 

.15442 

.98800 

.171641.98516 

7 

54 

.10279 

.99470 

.12014 

.99276 

.13744 

.99051 

.15471 

.98796 

.17193  I.98511 

6 

55 

.10308 

.99467 

.12043 

.99272 

.13773 

.99047 

.15500 

.98791 

.  17222  !.  98506 

5 

56 

.10337 

.99464 

.12071 

.99269 

.13802 

.99043 

.15529 

.98787 

.  17250  !.  98501 

4 

57 

10366 

.99461 

.12100 

..99265 

.13831 

.99039 

.15557 

.98782 

.17279I.98496 

3 

58 

.10395 

.99458 

.12129 

.99262 

.13860 

.99035 

.15586 

.98778 

.17308  .98491 

2 

59 

.10424 

.99455 

i  .12158 

.99258 

.13889 

.99031 

.15615 

.98773 

.17336 

.98486 

1 

60 

.10453 

.99452 

i.  12187 

.99255 

.13917 

.99027 

.15643 

.98769 

.17365 

.98481 

_0 

t 

Cofiiu 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 

84' 

83° 

82" 

81° 

80° 

TABLE   I.      SINES   AND   COSINES. 


A 

10° 

11°   1 

12° 

13°    1 

14° 

Sine  Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine  I  Cosin 

Sine 

Cosin 

, 

~o 

717365  '".98481 

.19081 

.98163 

.20791  .97815 

.22495  .97437 

.24192 

.97030 

60 

I 

.17393  .984761  .19109 

.98157 

.20820  .97809 

.22523  .97430 

.24220 

.97023 

59 

2 

.17422  .98471 

.19138 

.98152 

.20848  .97803 

.22552 

.97424 

.24249 

.97015 

58 

8 

.17451  .98466 

19167 

.98146 

.20877 

.97797 

.22580 

.97417 

.24277 

.97008 

57 

4 

.17479  .98461 

.  19195 

.98140 

.20905 

.97791 

.22608 

.97411 

.24305 

.97001 

56 

5  .17508  .98455 

.19224 

.98135 

.20933  .97784 

.22637 

.97404 

.24333 

.96994 

55 

6  .17537  .98450 

.19252 

.98129 

.209621.97778 

.22665 

/07398 

.24362 

.96987 

54 

7 

.17565  .98445 

.19281 

.98124 

.20990 

.97772 

.22693 

.97391 

.24390 

.96980  53 

8  .17594  .98440 

.19309 

.98118 

.21019 

.97766 

22722 

.97384 

.24418 

.96973 

52 

9  .17623  .98435 

.19338 

.98112 

.21047 

.97760 

.22750 

.97378 

.24446 

.96966 

51 

10  .17651  .98430 

.19366 

'.98107, 

.21076 

.97754 

.22778 

.97371 

.24474 

.96959 

50 

11  .17680 

.98425 

.19395 

.98101 

.21104 

.  97748  : 

.22807 

.97365 

j  .24503 

.96952 

49 

12  1.17708 

.98420 

.19423 

.98096 

.21132 

.97742 

.22835 

.97358  .24531 

.96945 

48 

13  .17737 

.98414 

.19452 

.98C90 

.21161 

.97735 

.22863 

.973511!  .24559 

.96937 

47 

14  .17766 

.98109 

.19481 

.98084 

.21189 

.97729 

.22892 

.97345  .24587 

.96930 

46 

15  '>,.  17794 

.98404 

.19509 

.98079 

.21218 

.97723 

.22920 

.97388  .24615 

.96923 

45 

16  '.17823 

.98399 

.19538 

.98073 

.21246 

.97717 

.22948 

.97331  .24644 

.96916 

44 

17  .17852 

.98394 

.19566 

.98067 

.21275 

.97711! 

.22977 

.97325 

.24672 

.96909 

43 

18  .17880 

.98389 

.19595 

.98061 

.21303 

.97705 

.23005 

.97318 

.24700 

.96902 

42 

19  .17909 

.98383 

.19623 

.98056 

.21331 

.97698 

.23033 

.97311 

.24728 

.96894 

41 

20  .17937 

.98378 

.19652 

.98050 

.21360 

.  97692  ; 

.23062 

.97304 

.24756 

.96887 

40 

21 

.17966 

.98373 

.19680 

.98044 

.21388 

.97686 

.23090 

.97298 

.24784 

.96880 

39 

22 

.17995 

.98368 

.19709 

.98039 

.21417 

.97680 

.23118 

.97291'  .24813 

.96873 

38 

23 

.18023 

.98362 

.19737 

.93033 

.21445 

.97673 

.23146 

.97284  ;  .24841 

.96866 

37 

24 

.18052 

.98357 

.19766 

.98027 

.21474 

.97667 

.23175 

.97278 

.24869 

.96858 

36 

25 

.18081 

.98352 

.19794 

.93021 

.21502 

.97661 

.23203 

.97271 

.24897 

.96851 

35 

26 

.18109 

.98347 

.19823 

.98016 

.21530 

.97655 

.23231 

.97264  .24925 

.96844 

34 

27 

.18138 

.98341 

.19851 

.93010 

.21559 

.97648 

.23260 

.97257,  .24954 

.96837 

33 

28 

.18166 

.98336 

.19880 

.93004 

.21587 

.97642 

.23288 

.972511  .24982 

.96829 

32 

29 

.18195 

.98331 

.19908 

.97998: 

.21616 

.97636 

.23316 

.97'244  .25010 

.96822 

31 

30 

.18224 

.98325 

.19937 

.97992  .21644 

.97630 

.23345 

.97237 

.25038 

.96815 

30 

31 

.18252 

.98320 

.19965 

.97987 

.21672 

.97623 

.23373 

.97230 

.25066 

.96807 

29 

32 

.18281 

.98315 

.19994 

.97981!  .21701 

.97617 

.23401 

.  97223  ! 

.25094 

.96800 

28 

33 

.18309 

.98310 

.20022 

.97975  .21729 

.97611 

.23429 

.97217!  .25122 

.96793 

27 

34 

.18338 

.98304 

.20051 

.97969  .21758 

.97604 

.23458 

.97210!  .25151 

.96786 

26 

35 

.18367 

.98299 

.20079 

.97963  .21786 

.97598 

.23486 

.97203i  .25179 

.96778 

25 

36 

.18395 

.98294 

.20108 

.97953  .21814 

.97592 

.23514 

.97196  .25207 

.  96771 

24 

37 

.18424 

.98288 

.20136 

.97952  .21843 

.97585 

.23542 

.97189!  .25235 

.96764 

23 

38 

.18452 

.98283 

.20165 

.97946  .21871 

.97579 

.23571 

.97182  .25263 

.96756 

22 

39 

.18481 

.98277 

.20193 

.97940 

.21899 

.97573 

.23599 

.97176  .25291 

.96749 

21 

40 

.18509 

.98272 

.20222 

.97934, 

.21928 

.97566, 

.23627 

.97169  .25320 

.96742 

20 

41 

.18538 

.98267 

.20250 

.97928 

.21956 

.97560 

.23656 

.97162  .25348 

.96734 

19 

42 

.18567 

.98261 

.20279 

.97922  .21985 

.97553 

.23684 

.97155  .25376 

.96727 

18 

43 

.18595 

.98256 

.2030? 

.97916  .22013 

.97547 

.23712 

.97148  |  .25404  .96719 

1  j 

44 

.186241.98250 

.20336 

.97910  .22041 

.97541 

.23740 

.  97141  j  .25432  .96712 

16 

45 

.18652  .98245 

.20364 

.97905  .22070 

.97534 

.23769 

.971341;  .25460 

.9G705 

15 

46 

.18681 

.98240 

.20393 

.97899 

.22098 

.97528; 

.23797 

.97127  .25488 

.96697 

14 

47 

.18710 

.98234 

.20421 

.97893 

.22126 

.97521 

.23825 

.97120  .25516  .96690 

13 

48 

.18738 

.98229!  1.20450 

.97887: 

.22155 

.97515 

.23853 

.97113  .25545  .90682 

12 

49 

.18767 

.98223 

.20478 

.97881 

.22183 

.975081 

.23882 

.97106  .25573  .96675 

11 

50 

.18795 

.98218 

.20507 

.97875 

.22212  .97502 

.23910 

.  97100  1  ,  .25601  '.96667 

10 

51 

.18824 

.98212 

.20535 

.97869 

.22240  '.97496 

.23938 

.97093  .25629  .96660 

9 

52 

.18852 

.98207 

.20563 

.97863 

.22268  .97489 

.23966 

.97086-  .25657  .96653 

8 

53 

.18881 

.98201 

.20592 

.97857 

.22297  .97483 

.23995 

.97079  .25685  .96645 

7 

54 

.18910 

.98196  .20620 

.97851 

.22325  .97476 

.24023 

.97072:  .25713  .96638 

6 

55 

.18938  .98190  .20649 

.  97845  :  !  .  22353  .  97470  .  24051 

.97065  .25741  .96630 

5 

56 

.18967  .98185 

.  20677  .  97839  .  22382  .  97463  .  24079 

.97058 

.25769  .96623 

4 

57 

.18995 

.98179 

.  20706  .  97833  .  22410  .  97457 

.24108  .97051 

..25798  .96615 

3 

58 

.19024 

.  981  74  .  20734  1  .  97827  .  22438  .  97450 

.24136  .97044 

.25826  .96608 

2 

59 

.19052  .981681  .20763  .  97821  ,  !.  22467  .97444 

.24164 

.97037 

.25854  .96600 

1 

60 

.19081  .98163 

.20791  .97815  .22495 

.97437 

.24192 

.97030 

.25882  .96593 

0 

Cosin  Sine 

Cosin  ;  Sine  :  Cosin 

Sine 

Cosin 

Sine 

Cosin  Sine 

79°   li   78°    I   77° 

76° 

75°   1 

TABLE   I.       SIXES   AND    COSINES. 


133 


1  — 

15°   ||.   16° 

17°    !    18° 

19° 

Sine 

Cosin 

Sine  Cosin 

Sine 

Cosin  1  Sine 

Cosin 

Sine 

Cosin 

1) 

.25882 

.96593 

.27564  .96126 

.29237 

.95630  .30902 

795106 

.32557 

.94552 

60 

1 

.25910 

.96585 

.27592  .96118 

.29265  .95622  .30929 

.95C37 

.32584 

.94542 

59 

2 

.25938 

.96578 

.27620i.96110 

.29293 

.95613!  .30957 

.95088 

.32612 

.94533 

58 

3 

.25966 

.96570 

.27648  .96102 

.29321 

.  95605  !  .30985 

.95079 

.32639 

.94523 

57 

4 

.259941.96562 

.27676  .96094 

.29348 

.95596  .31012 

.95070 

.32667 

.94514 

56 

5 

.26022  '.96555 

.27704  .96086 

.29376 

.95588!  .31040 

.95061 

.32694 

.94504 

55 

6 

.26050  .96547 

.27731  .96078 

.29404 

.95579 

.31068 

.95052 

.32722 

.94495 

54 

7 

.26079^.  96540 

.27759-96070  .29432 

.95571 

.31095 

.95043 

.32749 

.94485 

53 

8 

.  26107  ;.  96532 

.27787  .96062 

.29460 

.95562 

.31123 

.95033 

.32777 

.94476 

52 

9 

.  26135  !.  96524 

.27815  .96054 

.29487 

.95554 

.31151 

.95024 

.32804 

.944661  51 

10 

.26163 

.96517] 

.27843  .96046  .29515 

.95545 

.31178 

.95015 

.32832 

.94457 

50 

11 

.26191 

.96509 

.  27871  i.  96037  .29543 

.95536 

.31206 

.95006 

.32859 

.94447 

49 

12 

.26219  .96502 

.27899  .96029!  .29571 

.95528  .31233 

.94997 

.32887  .94438 

48 

13 

.26247  .96494 

.27927  .96021 

.29599 

.95519 

.312611.94988 

.32914  .94428 

47 

14 

.26275  .96486 

.27955  .96013 

.29626 

.95511 

.312891.94979 

.  32942  i.  94418 

46 

15 

.263031.96479 

.27983  .96005 

.29654 

.95502 

.31316  .94970 

.32969  .94409 

45 

16 

.26331  .96471 

.28011  .95997 

.29682 

.95493 

.31344  .94961 

.32997 

.94399 

44 

17 

.26359:.  96463 

.28039 

.95989 

.29710 

.95485 

.31372 

.94952 

.33024 

.94390 

43 

18 

.263871.96456 

.28067 

.95981 

.29737 

.95476 

.31399 

.94943 

.33051 

.94380 

42 

19 

.264151.96448 

.28095 

.95972 

.29765 

.95467! 

.31427 

.94933 

|  .33079 

.94370 

41 

20 

.26443 

.96440; 

.28123 

.95964 

.29793 

.95459:  .31454 

.94924 

.33106 

.94361 

40 

21 

.26471 

.96433; 

.28150 

.95956 

.29821 

.95450  i  .31482- 

.94915 

'i  .33134 

.94351 

39 

22 

.26500 

.96425 

.28178  .95948 

.29849 

.95441  1  .31510 

.94906 

i  .33161 

.94342 

38 

23 

.26528 

.964171 

.28206  .95940 

.29876 

.95433  .31537 

.94897 

!  .33189 

.94332 

37 

24 

.26556 

.96410 

.28234  .95931 

.29904 

.95424  |  .31565 

.94888 

.33216 

.94322 

36 

25 

.26584 

.96402 

.28262  .95923 

.29932 

.95415 

.31593 

.94878 

1  .33244 

.94313 

35 

26 

.26612 

.96394 

.28290:.  95915 

.29960 

.95407: 

.31620 

.94869 

.33271 

.94303 

34 

27 

.26640 

.96386 

.28318  .95907 

.29987 

.95398  .31648 

•.94860 

.33298 

.94293 

33 

28 

.26668 

.96379 

.28346 

.95898 

.30015 

.95389 

.31675 

.94851 

.33326 

.94284 

32 

29 

.26696 

.96371! 

.28374 

.95890 

.30043 

.95380 

.31703 

.94842 

.33353 

.94274 

31 

30 

.26724 

.96363 

.28402 

.95882 

.30071 

.95372 

.31730 

.94832 

.33381 

.94264 

30 

31 

.26752 

.96355 

.28429 

.95874 

.30098 

.95363 

.31758 

.94823 

.33408 

.94254 

29 

32 

.26780 

.96347 

.28457 

.95865 

.30126 

.95354 

.31786 

.94814 

.33436 

.94245 

28 

33 

.26808 

.96340 

.28485 

.95857 

.30154 

.95345 

.31813 

.94805 

-33463 

.94235 

27 

34 

.26836 

.96332 

.28513 

.95849 

.30182 

.95337 

.31841 

.94795 

.33490 

.94225 

26 

35 

.26864 

.96324 

.28541 

.95841 

.30209 

.95328 

.31868 

.94786 

.33518 

.94215 

25 

.36 

.26892 

.96316 

.28569 

.95832 

.30237 

.95319 

.31896 

.94777 

.33545 

.94206 

24 

37 

.26920 

.96308 

.28597 

.95824 

.30265 

.95310 

.31923 

.94768 

.33573 

.94196 

23 

38 

.26948 

.96301 

.28625 

.95816 

.30292 

.95301 

.31951 

.94758 

.33600 

.94186 

22 

39 

.26976 

.96293 

.28652  .95807 

.30320 

.95290 

.31979 

.94749 

.33627 

.94176 

21 

40 

.27004 

.96285.! 

.28680  .95799 

.30348 

.95284 

.32006 

.94740 

.33655 

.94167 

20 

41 

.27032 

.96277 

.28708  .95791 

.30376 

.95275 

.32034 

.94730 

.33682 

.94157 

19 

42 

.27060 

.96269 

.28736 

.95782 

.30403 

.95266 

.32061 

.94721 

.33710 

.94147 

18 

43 

.27088 

.96261 

.28764 

.95774 

.30431 

.95257 

.32089 

.94712 

.33737 

.94137 

17 

44 

.27116 

.96253 

.28792  .95766 

.30459 

.95248 

.32116 

.94702 

.33764 

.94127 

16 

45 

.27144 

.96246 

28820 

.95757 

.30486 

.95240 

.32144 

.94693 

.33792 

.94118 

15 

46 

.27172 

.96238 

'28847 

.95749 

.30514 

.95231 

.32171 

.94684 

.33819 

.94108 

14 

47 

.27200 

.96230 

.'28875 

.95740 

.30542 

.95222 

.32199 

.94674 

.33846 

.94098 

13 

48 

.27228 

.96222 

.28903 

.95732 

.30570 

.95213 

.32227 

.94665 

.33874 

.94088 

12 

49 

.27256 

.96214! 

.28931 

.95724 

.30597 

.95204 

.32254 

.94656 

33901 

.94078 

11 

50 

.27284 

.  96206  | 

.28959 

.95715 

.30625 

.95195 

.32282 

.94646 

.33929 

.94068 

10 

51 

.27312 

.96198 

.28987 

.95707 

.30653 

.95186 

.32309 

.94637 

.33956 

.94058 

9 

52 

.27340 

.96190 

.29015 

.95698 

.30680 

.951771 

.32337 

.94627 

1  .33983 

.94049 

8 

53 

.27368 

.96182 

.290421.95690 

.30708 

.95168 

.32364 

.94618 

.84011 

.94039 

7 

54 

.27396 

.96174 

.29070 

.95681 

.30736 

.95159: 

.32392 

.94609 

i.  34038 

.94029 

6 

55 

.27424 

.96166! 

.29098 

.  95673 

'  .30763 

.95150: 

.32419 

.94599 

.34065 

.94019 

5 

56 

.27452 

.96158 

.29126 

.95664 

.30791 

.95142 

.32447 

.94590 

.34093 

.94009 

4 

57 

.27480 

.96150 

.29154  .95656 

.30819 

.95133 

.32474 

.94580 

.34120 

.93999 

3 

58 

.27508 

.96142 

.29182 

.95647 

.30846 

.95124! 

.32502 

.94571 

.34147 

.93989 

2 

59 

.27536 

.96134 

.29209 

.95639 

.30874 

.95115 

.32529 

.94561 

.34175 

.93979 

1 

60 

.27564 

.96126 

.29237 

.95630 

.30902 

.95106 

.32557 

.94552 

.34202 

.93969 

0 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  i 

Cosin 

Sine 

Cosin 

Sine 

74° 

73° 

72° 

71° 

70° 

134 


TABLE  I.      SINES   AND   COSINES. 


20° 

21°    | 

22* 

23°   | 

24° 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  Cosin 

0 

.34202 

.93969 

.35837 

.93358 

.37461 

.92718 

.39073 

92050 

.  40674  !.  91355 

60 

1 

.34229 

.93959 

.35864 

.93348 

.37488 

.92707 

.39100 

92039 

.40700  .91343 

59 

2 

.34257 

.93949 

.35891 

.93337 

.37515 

.92697 

.39127 

92028 

.40727 

.91331 

58 

3 

.34284 

.93939 

.35918 

.93327 

.37542 

.92686 

.39153 

92016 

.40753 

.91319 

57 

4 

.34311 

.93929 

.35945 

.93316 

.37569 

.92675 

.39180 

92005 

.40780 

.91307 

56 

5 

.34339 

.93919 

.35973 

.93306 

.37595 

.92664 

.39207 

91994 

.40806 

.91295 

55 

6 

.34366 

.93909 

.36000 

.93295 

.37622 

.92653 

.39234 

91982 

.40833 

.91283 

54 

7 

.34393 

.93899 

.36027 

.93285 

.37649 

.92642 

.39260 

91971 

.40860 

.912721  53 

6 

.34421 

.93889 

.36054 

.93274 

.37676 

.92631 

.39287 

91959 

.40886 

.91260 

52 

9 

.34448 

.93879 

.36081 

.93264 

.37703 

.92620 

.39314 

91948 

.40913 

.91248 

51 

10 

.34475 

.93869 

.36108 

.93253 

.37730 

.92609 

.39341 

91936 

.40939 

.91236 

50 

11 

.34503 

.93859 

.36135 

.93243 

.37757 

.92598 

.39367 

.91925 

.40966 

.91224 

49 

12 

.34530 

.93849 

.36162  '.93232 

.37784 

.92587 

.39394 

.91914 

.40992 

.91212  48 

13 

.34557 

.93839 

.36190 

.93222 

.37811 

.92576 

.39421 

.91902 

.41019 

.91200  47 

14 

.34584 

.93829 

.36217 

.93211 

.37838 

.92565 

.39448 

.91891 

.41045 

.91188  46 

15 

.34612 

.93819 

.36244 

.93201 

.37865 

.92554 

.39474 

.91879 

.41072 

.91176 

46 

16 

.34639 

.93809 

.36271 

.93190 

.37892 

.92543 

.39501 

.91868 

.41098 

.91164 

44 

17 

.34666 

.93799 

.36298 

.93180 

.37919 

.92532 

.39528 

.91856 

.41125 

.91152  43 

18 

.34694 

.93789 

.36325 

.93169 

.37946 

.92521 

.39555 

.91845 

.41151 

.91140 

42 

19 

.34721 

.93779 

.36352 

.93159 

.37973 

.92510 

.39581 

.91833 

.41178 

.91128 

41 

20 

.34748 

.93769 

.36379 

.93148 

.37999 

.92499 

.39608 

.91822 

.41204 

.91116 

40 

21 

.34775 

.93750 

.36406 

.93137 

.38026 

.92488 

.39635 

.91810 

.41231 

.91104 

39 

22 

.34803 

.93748 

.36434 

.93127 

.38053 

.92477 

.39661 

.91799 

.41257 

.91092 

38 

23 

.34830 

.93738 

.36461 

.93116 

.38080 

.92466 

.39688 

.91787 

.41284 

.91080 

37 

24 

.34857 

.93728 

.36488 

.93106 

.38107 

.92455 

.39715 

.91775 

.41310 

.91068 

36 

25  .34884 

.93718 

.36515 

.93095 

.38134 

.92444 

.39741 

.91764 

.41337 

.91056  35 

26  .34912 

.93708 

.36542 

.93084 

.38161 

.92432 

.39768 

.91752 

.41363 

.91044 

34 

27  .34939 

.93698 

.36569 

.93074 

.38188 

.92421 

.39795 

.91741 

.41390 

.91032 

33 

28 

.34966 

.93688 

.36596 

.93063 

.38215  .92410 

.39822 

.91729 

.41416 

.91020 

32 

29 

.34993 

.93677 

.36623 

.93052 

.88241 

.92399 

.39846 

.91718 

.41443 

.91008 

31 

30 

.35021 

.93667 

.36650 

.93042 

.38268 

.92388 

.39875 

.91706 

.41469 

.90996 

30 

31 

.35048 

.9365? 

.36677 

.93031 

.38295 

.92377 

.39902 

.91694 

.41496 

.90984 

29 

32 

.35075 

.93647 

.36704 

.93020 

.38322 

.92366 

.39928 

.91688 

.41522 

.90972 

28 

33 

.35102 

.93637 

.36731 

.93010 

.38349 

.92355 

.39955 

.91671 

.41549 

.90960 

27 

34 

.35130 

.93626 

.36758 

.92999 

.38376 

.92343 

.39982 

.91660 

.41575 

.90948 

26 

35 

.35157 

.93616 

.36785 

.92988 

.38403 

.92332 

.40008 

.91648 

.41602 

.90936 

25 

36 

.35184 

.93606 

.36812 

.92978 

.38430 

.92321 

.40035 

.91636 

.41628 

.90924 

24 

37 

.35211 

.93596 

.36839 

.92967 

.38456 

.92310 

.40062 

.91625 

.41655 

.90911 

23 

38 

.35239 

.93585 

.36867 

.92956 

.38483 

.92299 

.40088 

.91613 

.41681 

.90899 

22 

39 

.35266 

.93575 

.36894 

.92945 

.38510 

.92287 

.40115 

.91601 

.41707 

.90887 

21 

40 

.35293 

.93565 

.36921 

.92935 

.38537 

.92276 

.40141 

.91590 

.41734 

.90875 

20 

41 

.35320 

.93555 

.36948 

.92924 

.38564 

.92265 

.40168 

.91578 

.41760 

.90863 

19 

42 

.35347 

.93544 

.36975  .92913 

.38591 

.92254 

.40195 

.91566 

.41787 

.90851 

18 

43 

.35375 

.93534 

.37002 

.92902 

.38617 

.92243 

.40221 

.91555 

.41813 

.90839 

17 

44 

.35402 

.93524 

.37029 

.92892 

.38644 

.92231 

.40248 

.91543 

.41840 

.90826 

16 

45 

.35429 

.93514 

.370561.92881 

.38671 

.92220 

.40275 

.91531 

.41866 

.90814 

15 

46 

.35456 

.93503 

.37083  .92870 

.38698 

.92209 

.40301 

.91519 

.41892 

.90802 

14 

47 

.35484 

.93493 

.37110 

.92859 

.38725 

.92198 

.40328 

.91508 

.41919 

.90790 

13 

48 

.35511 

.93483 

.37137 

.92849 

.38752 

.92186 

.40355 

.91496 

.41945 

.90778 

12 

49 

.35538 

.93472 

.37164 

.92838 

.38778 

.92175 

.40381 

.91484 

.41972 

.90766  11 

50 

.35565 

.934*62 

.37191 

.92827 

.38805 

.92164 

.40408  .91472 

.41998 

.90753 

10 

51 

.35592 

.93452 

.37218 

.92816 

.38832 

.92152 

.40434  .91461 

.42024 

.90741 

9 

52 

.35619 

.93441 

.37245 

.92805 

.38859 

.92141 

.  40461  !.  91449 

.42051 

.90729 

8 

53 

.35647 

.93431 

.37272 

.92794 

.38886 

.92130 

.404881.91437 

.42077 

.90717 

7 

54 

.35674 

.93420 

.37299 

.92784 

.38912 

.92119 

.40514  .91425 

.42104 

.90704 

6 

55 

.35701 

.93410 

.37326 

.92773 

.38939 

.92107 

.40541  .91414 

.42130 

.90692 

5 

56 

.35728 

.93400 

.37353 

.92762 

.38966 

.92096 

.405671.91402 

.42156 

.90680 

4 

57 

.35755 

.93389 

.37380 

.92751 

.38993 

.92085 

.  40594  ;.  91  390 

.42183 

.90668 

3 

58 

.35782 

.93379 

.37407 

.92740 

.39020 

.92073 

.40621  1.91378 

.42209 

.90655 

2 

59 

.35810 

.93368 

.37434 

.92729 

.39046 

.92062 

.40647  .91366 

.42235 

.90643 

1 

60 

.35837 

.93358 

.37461 

.92718 

.39073 

.92050 

.40674  .91355 

,42262 

.90631 

0 

t 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  ]  Sine 

Cosin 

Sine 

f 

69° 

68° 

67° 

66° 

65' 

TABLE   I.      SINES  AND   COSINES. 


25° 

26° 

27° 

28° 

29° 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

"o 

.42262 

.90631 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

60 

1 

.42288 

.90618 

.43863 

.89867 

.45425 

.89087 

.46973 

.88281 

.48506 

.87448 

59 

2 

.42315 

.90606 

.43889 

.89854 

.45451 

.89074 

.46999 

.88267 

.48532 

.87434 

58 

3 

.42341 

.90594 

.43916 

,89841 

.45477 

.89061 

.47024 

.88254 

.48557 

.87420 

57 

4 

.42367 

.90582 

.43942 

.89828 

.45503 

.89048 

.47050 

.88240 

.48583 

.87406 

56 

5 

.42394 

.90569 

.43968 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608 

.87391 

55 

6 

.42420 

.90557 

.43994 

.89803 

.45554 

.89021 

,47101 

.88213 

.48634 

.87377 

54 

7 

.42446 

.90545 

.44020 

.89790 

.45580 

.89008 

.47127 

.88199 

.48659 

.87363 

53 

8 

.42473 

.90532 

.44046 

.89777 

.45606 

.88995 

.47153 

.88185 

.48684 

.87349 

52 

9 

.42499 

.90520 

.44072 

.89764 

.45632 

.88981 

.47178 

.88172 

.48710 

.87335 

51 

10 

.42525 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

.42552 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

.42578 

.90483 

.44151 

.89726 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88928 

.47281 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90408 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48938 

.87207 

42 

19 

.42762 

.90396 

.44333 

.89636 

.45891 

.88848 

.47434 

.88034 

.48964 

.87193 

41 

20 

.42788 

.90383 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

.44385 

.89610 

.45942 

.88822 

.47486 

.88006 

.49014 

.87164 

39 

22 

.42841 

.90358 

.44411 

.89597 

.45968 

.88808 

.47511 

.87993 

.49040 

.87150 

38 

23X42867 

.90346 

.44437 

.89584 

.45994 

.88795 

.47537 

.87979 

.49065 

.87136 

37 

24^.42894 

.90334 

.44464 

.89571 

.46020 

.88782 

.47562 

.87965 

.49090 

.87121 

36 

25  .42920 

.90321 

.44490 

.89558 

.46046 

.88768 

.47588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.46072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.44542 

.89532 

.46097 

.88741 

.47639 

.87923 

.49166 

.87079 

33 

28 

.42999 

.90284 

.44568 

.89519 

.46123 

.88728 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.90271 

.44594 

.89506 

.46149 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

.43077 

.90246 

.44646 

.89480 

.46201 

.88688 

.47741 

.87868 

.49268 

.87021 

29 

32 

.43104 

.90233 

.44672 

.89467 

.46226 

.88674 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130 

.90221 

.44698 

.89454 

.46252 

.88661 

.47793 

.87840 

.49318 

.86993 

27 

34 

.43156 

.90208 

.44724 

.89441 

.46278 

.88647 

.47818 

.87826 

.49344 

.86978 

26 

35 

.43182 

.90196 

.44750 

.89428 

.46304 

.88634 

.47844 

.87812 

.49369 

.86964 

25 

36 

.43209 

.90183 

.44776 

.89415 

.46330 

.88620 

.47869 

.87798 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88607 

.47895 

.87784 

.49419 

.86935 

23 

38 

.43261 

.90158 

.44828 

.89389 

.46381 

.88593 

.47920 

.87770 

.49445 

.86921 

22 

39 

.43287 

.90146 

.44854 

.89376 

.46407 

.88580 

.47946 

.87756 

.49470 

.86906 

21 

40 

.43313 

.90133 

.44880 

.89363 

.46433 

.88566 

.47971 

.87743 

.49495 

.86892 

20 

41 

.43340 

.90120 

.44906 

.89350 

.46458 

.88553 

.47997 

.87729 

.49521 

.86878 

19 

42 

.43366 

.90108 

.44932 

.89337 

.46484 

.88539 

.48022 

.87715 

.49546 

.86863 

18 

43 

.43392 

.90095 

.44958 

.89324 

.46510 

.88526 

.48048 

.87701 

.49571 

.86849 

17 

44 

.43418 

.90082 

.44984 

.89311 

.46536 

.88512 

.48073 

.87687 

.49596 

.86834 

16 

45 

.43445 

.90070 

.45010 

.89298 

.46561 

.88499 

.48099 

.87673 

.49622 

.86820 

15 

46 

.43471 

.90057 

.45036 

.89285 

.46587 

.88485 

.48124 

.87659 

.49647 

.86805 

14 

47 

.43497 

.90045 

.45062 

.89272 

.46613 

.88472 

.48150 

.87645 

.49672 

.86791 

13 

48 

.43523 

.90032 

.45088 

.89259 

.46639 

.88458 

.48175 

.87631 

.49697 

.86777 

12 

49 

.43549 

.90019 

.45114 

.89245 

.46664 

.88445 

.48201 

.87617 

.49723 

.86762 

11 

50 

.43575 

.90007 

.45140 

.89232 

.46690 

.88431 

.48226 

.87603 

.49748 

.86748 

10 

51 

.43602 

.89994 

.45166 

.89219 

.46716 

.88417 

.48252 

.87589 

.49773 

.86733 

9 

52 

.43628 

.89981 

.45192 

.89206 

.46742 

.88404 

.48277 

.87575 

.49798 

.86719 

8 

53 

.43654 

.89968 

.45218 

.89193 

.46767 

.88390 

.48303 

.87561 

.49824 

.86704 

7 

54 

.43680 

.89956 

.45243 

.89180 

.46793 

.88377 

.48328 

.87546 

.49849 

.86690 

6 

55 

.43706 

.89943 

.45269 

.89167 

.46819 

.88363 

.48354 

.87532 

.49874 

.86675 

5 

56 

.43733 

.89930 

.45295 

.89153 

.46844 

.88349 

.48379 

.87518 

.49899 

.86661 

4 

57 

.43759 

.89918 

.45321 

.89140 

.46870 

.88336 

.48405 

.87504 

.49924 

.86646 

3 

58 

.43785 

.89905 

.45347 

.89127 

.46896 

.88322 

.48430 

.87490 

.49950 

.86632 

2 

59 

.43811 

.89892 

.45373 

.89114 

.46921 

.88308 

.48456 

.87476 

.49975 

.86617 

1 

60 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

.50000 

.86603 

J) 

t 

Cosin 

Sine 

Cosin 

Sine 

Cosin  j  Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 

64° 

63° 

62° 

61° 

60° 

TABLE   I.      SINES   AND   COSINES. 


SO9 

31°   |    32° 

33° 

34° 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  1  Cosin 

Sine 

Cosin 

f 

~o 

.50000 

.86603 

.51504 

785717 

.52992 

.84805 

.54464  .83867 

.55919 

.82904 

60 

1 

.50025 

.86588 

..51529 

.85702 

.53017 

.84789 

.  54488  ;.  83851 

.55943 

.82887 

59 

2 

.50050 

.86573 

.51554 

.85687 

.53041 

.84774 

.545131.83835 

.55968 

.82871 

58 

3 

.50076 

.86559 

.51579 

.85672 

.53006  .84759  .54537  .83819 

.55992 

.82855 

57 

4 

.50101 

.86544 

.51604 

.85657 

.53091  .84743  .54561  .83804 

.56016 

.82839 

56 

5 

.50126 

.86530 

.51628 

.85642 

.53115  .84728  .54586  .83788 

.56040 

.82822 

55 

6 

.50151 

.86515 

.51653 

-85627 

.53140  .84712 

.54610;.  83772 

.56064 

.82806 

54 

7 

.50176 

.86501 

.51678 

.85612 

.531641.84697 

.54635  .83756 

.56088 

.82790 

53 

8 

.50201 

.86486 

.51703 

.85597 

.53189  .84681' 

.54659  .83740 

.56112 

.82773 

52 

9 

.50227 

.86471 

.51728 

.85582 

.53214 

.84666 

.54683  '.83724 

.56136 

.827571  51 

10 

.50252 

.86457 

.51753 

.85567 

.53238 

.  84650  f 

.54708  .83708 

.56160 

.82741 

50 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.  84635  i 

.547321.83692 

.56184 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.53288 

.84619 

.54756 

.83076  .50208 

.82708 

48 

13 

.50327 

.86413 

.51828 

.85521 

,53312  .84604 

.54781 

.83660 

.50232 

.82692 

47 

14 

.50352 

.86398 

.51852 

.85506 

.53337  .84588! 

.54805 

.83045 

.50256 

.82675 

46 

15 

.50377 

.86384 

.51877 

.85491 

.53361 

.84573 

.54829 

.83629 

.56280 

.82659 

45 

16 

.50403 

.86369 

.51902 

.85476 

.53386 

,84557 

.54854  .83613 

.56305 

.82643 

44 

17 

.50428 

.86354 

.51927 

.85461 

.53411 

.84542 

.54878  .83597 

.56329 

.82626 

43 

18 

.50453 

.86340 

.51952 

.85446 

.53435  81526) 

.54902  .83581 

.56353 

.82610 

42 

19 

.50478 

.86325 

.51977 

.85431 

.534601.  8451  li 

.54927  .83565 

.56377 

.82593 

41 

20 

.50503 

.86310 

.52002 

.85416 

.53484 

.84495;  .54951  .83549 

.56401 

.82577 

40 

21 

.50528 

.86295 

.52026 

.85401 

.53509 

.84480  .54975  '.83533 

.56425 

.82561 

39 

22 

.50553 

.86281 

.52051 

.85385 

.53534  .84464   54999k  8351  7 

.56449 

.82544 

38 

23 

.50578 

.86266 

!  .52076 

.  85370  | 

.53558  .84448 

.55024k83501 

.56473 

.8252*37 

24 

.50603 

.86251 

.52101 

.85355 

.53583  .84433 

.55048  .83485 

.56497 

.82511 

i  38 

25 

.50628 

.86237 

.52126 

.85340 

.53607 

.84417 

.55072  ',83469 

.56521 

.82495 

35 

26 

.50654 

.86222 

.52151 

.85325 

.53632 

.84402 

.55097  .83453 

.56545 

.82478 

34 

27 

.50679 

.86207 

.52175 

.85310 

.53656 

.84386 

.55121  .83437 

.56569 

.82462 

33 

28 

.50704 

.86192 

.52200 

.85294 

.53681 

.84370 

.55145  .83421 

,56593 

.82446 

|32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84355 

.551691.83405 

.56617 

.82429 

!  31 

30 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339 

.55194 

.83389 

56641 

,82413 

30 

31 

.50779 

.86148 

.52275 

.85249 

.53754 

.84324! 

.55218 

.83373  ,56665 

.82396 

29 

32 

.50804 

.86133 

.52299 

.85234 

.53779 

.84308 

.55242 

.83356 

.56689 

82380 

28 

33 

.50829 

.86119 

.52324 

.85218 

.53804 

.84292 

.55266  .83340 

.56713 

.82363 

i  27 

34 

.50854 

.86104 

.52349 

.85203] 

.53828 

.84277 

.552911.83324 

.56736 

.82347 

'  26 

35 

.50879 

.86089 

.52374 

.85188 

.53853  .84261! 

.55315  .83308 

.56760 

.82330 

25 

36 

.50904 

.86074 

.52399 

.85173 

.53877  .84245 

.55339  .83292 

.56784 

.82314 

24 

37 

.50929 

.86059 

.52423 

.851571 

.53902  .84230 

.55363  .83276  .56808 

.82297 

23 

38 

.50954 

.86045 

.52448 

.85142 

.  53926  i.  84214 

.55388  .83260 

.56832 

.82281 

22 

39 

.50979 

.86030 

.52473 

.85127 

.539511.84198 

.55412  .83244 

.56856 

.82204 

21 

40 

.51004 

86015 

.52498 

.85112- 

.53975  .84182 

.55436  .  83228  (  .56880 

.82248 

,  20 

41 

.51029 

.86000 

.52522 

.85096' 

.54000 

.84167 

.55460  .  83212  '  .56904 

.82231 

19 

42 

.51054 

.85985 

.52547 

.85081 

.540241.84151 

.55484  .83195  .56928 

.82214 

18 

43 

.51079 

.85970 

.52572 

.85066 

.54049 

.84135 

.55509  .83179  .56952 

.82198 

17 

44 

.51104 

.85956 

.52597 

.85051  ! 

.54073 

.84120 

.5b533!.  83163  .50976 

.82181 

16 

45 

.51129 

.85941 

.52621 

.85035 

.54097 

.84104 

.55557  .83147  .57000 

.82165 

15 

46 

.51154 

.85926 

.52646 

.85020: 

.54122 

.84088 

.55581 

.83131  i  .57024 

.82148 

14 

47 

.51179 

.85911 

.53671 

.85005! 

.54146 

.84072 

.55605 

.83115  .57047 

.82132 

13 

48 

.51204 

.85896 

.52696 

.849891 

.54171 

.84057 

.55630 

.83098  .57071 

.82115 

12 

49 

.512291.85881 

.52720 

.849741 

.54195 

.84041 

.55654 

.83082  .57095 

.82098 

11 

50 

.5125'  .85866 

.52745 

.84959 

.54220  .84025 

.55678 

.83066  .57119 

.82082 

10 

51 

.51279 

.85851 

.52770 

.84943 

.54244 

.84009 

.55702 

.83050  .57113 

.82065 

9 

52 

.  51304  L85&36 

.52794 

.84928 

.54269 

.83994 

.55726 

.83034  .57107 

.82048 

8 

53 

.51329 

.85821 

.52819 

.84913 

.54293 

.83978 

.55750 

.83017 

.57191 

.82032 

,  7 

54 

.51354 

.85806 

.52844 

.84897) 

.54317 

.83962 

.55775 

.83001 

.57215 

.82015 

6 

55 

.51379 

.85792 

.52869 

.84882 

.54342 

.83946 

.55799 

.82985 

.57238 

.81999 

5 

56 

.51404 

.85777 

.52893 

.84866 

.54366 

.83930 

.55823 

.82969 

.57262 

.81982 

4 

57 

.51429  .85762 

.52918 

.84851; 

.54391 

.83915 

.55847 

.82953 

.57286 

.81965 

3 

58 

.51454L85747 

.52943 

.84836 

.54415 

.83899 

.55871 

.82936 

.57310 

.81949 

2 

59 

.51479 

.85732 

.52967 

.84820 

.54440  .83883 

.55895 

.82920 

.57334 

.81932 

1 

52 

.51504 

.85717 

.52992 

.84805' 

.544641.83867 

.55919 

.82904 

.57358 

.81915 

0 

/ 

Cosin 

Sine 

Cosin  |  Sine 

Cosin  |  Sine 

Cosin 

Sine 

Cosin 

Sine 

f 

59° 

58° 

57° 

56° 

55° 

TABLE   I.      SINES   AND    COSINES. 


137 


35° 

36° 

37° 

38° 

39° 

7 

Sine  Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

~0  .57358 

.81915 

.58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62932 

.77715 

60 

1  !  .57381 

.81899 

.58802 

.80885 

.60205 

.79846 

.61589 

.78783 

.62955 

.77696 

59 

2 

.57405 

.81882 

.58826 

.80867 

.60228 

.79829 

.61612 

.78765 

.62977 

.77678 

58 

3 

.57429 

.81865 

.58849 

.80850 

.60251 

.79811 

.61635 

.78747 

.63000 

.77660 

57 

4 

.57453 

.81848 

.58873 

.80833 

.60274 

.79793 

.61658 

.78729 

.63022 

.77641 

56 

5 

.57477 

.81832! 

.58896 

.80816 

.60298 

.79776 

.61681 

.78711 

.63045 

.77623 

55 

6 

.57501 

.81815 

.58920 

.80799 

.60321 

.79758 

.61704 

.78694 

.63068 

.77605 

54 

7 

.57524 

.81798 

.58943 

.80782 

.60344 

.79741 

.61726 

.78676 

.63090 

.77586 

53 

8 

.57548  .81782 

.58967 

.80765 

.60367 

.79723 

.61749 

.78658 

.63113 

.77568 

52 

9  .57572 

.81765 

.58990 

.80748 

.60390 

.79706 

.61772 

.78640 

.63135 

.77550 

51 

10 

.57596 

.81748 

.59014 

.80730 

.60414 

.79688 

.61795 

.78622 

.63158 

.77531 

50 

11 

.57619 

.81731 

.59037 

.80713 

.60437 

.79671 

.61818 

.78604 

.63180 

.77513 

40 

12 

.57643 

.81714 

.59061 

.80696 

.60460 

.79653 

.61841 

.78586 

.63203 

.77494 

48 

13  i  .57667 

.81698 

.59084 

.80679 

.60483 

.79635 

.G1864 

.78568! 

.63225 

.77476 

47 

14  1  .57691 

.81681 

.59108 

.80662 

.60506 

.79618 

.61887 

.78550; 

.63248 

.77458 

46 

15  !  .57715 

.81664 

.59131 

.80644 

.60529 

.79600 

.61909 

.78532 

.63271 

.77439 

45 

16  !  .57738 

.81647 

.59154 

.80627 

.60553 

.79583 

.61932 

.78514; 

.63293 

.77421 

44 

17  .57762 

.81631 

.59178 

.80610 

.60576 

.79565 

.61955 

.78496 

.633161.77402 

43 

18  !  .57786 

.81614 

.59201 

.80593 

.60599 

.79547 

.61978 

.78478 

.63338!.  77384 

42 

19  !  .57810 

.81597 

.59225 

.80576 

.60622 

.79530 

.62001 

.78460! 

.633611.77366 

41 

20 

.57833 

.81580 

.59248 

.80558 

.60645 

.79512 

.62024 

.78442! 

.63383 

.77347 

40 

21 

.57857 

.81563 

.59272 

.80541 

.60668 

.79494 

.62046 

.78424 

.63406 

.77329 

39 

22  .57881 

.81546 

.59295 

.80524 

.60691 

.79477 

.62069  '.78405 

.63428 

.77310 

38 

23  .57904 

.81530 

.59318 

.80507 

.60714 

.79459 

.62092;.  7  8387 

.63451 

.77292 

37 

24  I  .57928 

.81513 

.59342 

.80489 

.60738 

.79441 

.621151.78369 

.63473 

.77273 

36 

25  !  .57952 

.81496 

.59365 

.80472 

.60761 

.79424 

.62138 

.78351 

.634961.77255 

35 

26 

.57976 

.81479 

.59389 

.80455 

.60784 

.79406 

!  .62160 

.  78333  ! 

.635181.77236 

34 

27 

.57999 

.81462 

.59412 

.80438 

.60807 

.79388 

.62183  .78315 

.63540|.  77218 

33 

28 

.58023 

.81445 

.59436 

.80420 

.60830 

.79371 

.62206  .78297 

.63563!.  77199 

32 

29 

.58047 

.81428 

.59459  .80403 

60853 

.79353 

.62229 

.78279 

.63585 

.77181 

31 

30 

.58070 

.81412 

.59482 

.80386 

.60876 

.79335 

.62251 

.78261 

.63608 

.77162 

30 

31 

.58094 

.81395 

.59506 

.80368 

.60899 

.79318 

.62274 

.78243 

.63630 

.77144 

29 

32 

.58118 

.81378 

.59529 

.80351 

.60922 

.79300 

.62297 

.78225 

.63653 

.77125 

28 

33 

.58141 

.81361 

.59552 

.80334 

.60945 

.79282 

.62320 

.78206 

.63675 

.77107 

27 

34 

.58165 

.81344 

.595761.80316 

.60968 

.79264  |  .62342 

.78188 

.63698 

.77088 

26 

35 

.58189 

.81327 

.59599 

.80299 

.60991 

.79247 

!  .62365 

.78170 

.63720 

.77070 

25 

36 

.58212 

.81310 

.59622 

.80282 

.61015 

.79229 

.62388 

.78152 

.63742 

.77051 

24 

37 

.58236 

.B1293 

.59646 

.80264 

.61038 

.79211 

.62411 

.78134 

.63765 

.77033 

23 

38 

.58260 

.81276 

.59669 

.80247 

.61061 

.79193 

.62433 

.78116 

.63787 

.77014 

22 

39 

.58283 

.81259 

.59693 

.80230 

.61084 

.79176 

.62456 

.78098 

.63810 

.76996 

21 

40 

.58307 

.81242 

.59716 

.80212 

.61107 

.79158 

.62479 

.78079 

.63832 

.76977 

20 

41 

.58330 

.81225 

.59739 

.80195 

.61130 

.79140 

.62502 

.78061 

.63854 

.76959 

19 

42 

.58354 

.81208 

.59763 

.80178 

.61153 

.79122 

.62524 

.78043 

.63877 

.76940 

18 

43 

.58378 

.81191 

.59786 

.80160 

.61176 

.79105 

.62547 

.78025 

.63899 

.76921  17 

44 

.58401 

.81174 

.59809 

.80143 

.61199 

.79087 

.62570 

.78007 

.63922 

.76903!  16 

45 

.58425 

.81157 

.59832 

.80125 

.61222 

.79069 

.62592 

.77988 

.63944 

.76884  15 

48 

.58449 

.81140 

.59856 

.80108 

.61245 

.79051 

.62615 

.77970 

.63966  .76866!  14 

47 

.58472 

.81123 

.59879 

.80091 

.61268 

.79033 

.62638 

.77952 

.63989  .76847  13 

48 

.58496 

.81106 

.59902 

.80073 

.61291 

.79016 

.62660 

.77934 

.64011 

.76828 

12 

49 

.58519 

.81089 

.59926 

.80056 

.61314 

.78998 

.62683 

.77916 

.64033 

.76810 

11 

50 

.58543 

.81072 

.59949 

.80038 

.61337 

.78980 

.62706 

.77897 

.64056 

.76791 

10 

51 

.58567 

.81055 

.59972 

.80021 

.61360 

.78962 

.62728 

.77879 

.64078 

.76772 

9 

52 

.58590 

.81038 

.59995 

80003 

.61383 

.78944 

.62751 

.77861 

.64100 

.76754 

8 

53 

.58614 

.81021 

.60019 

.79986 

.61406 

.78926 

.62774 

.77843; 

.64123 

.76735 

7 

54 

.58637 

.81004 

.60042 

.79968 

.61429 

.78908 

.62796 

.77824 

.64145 

.70717 

6 

55 

.58661 

.80987 

.60065 

.79951 

.61451 

.78891 

.62819 

.77806 

.64167 

.76698 

5 

56 

.58684 

.80970 

.60089 

.79934 

.61474 

.78873 

.62842 

.77788 

.64190 

.76679 

4 

57 

.58708 

.80953 

.60112 

.79916 

.61497 

.78855 

.62864 

.77769 

.64212 

.76661 

3 

58 

.58731 

.80936 

.60135 

.79899 

.61520 

.78837 

.62887 

.77751 

.642341.76642 

2 

59 

.58755 

.80919 

.60158 

.79881 

.61543 

.78819 

.62909 

.77733 

.64256 

.76623 

1 

60  .58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62932 

.77715 

.64279 

.76604 

J) 

f 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  1  Sine 

/ 

54° 

53° 

52° 

51° 

50° 

TABLE  I.      SINES  AND   COSINES. 


40° 

41° 

42° 

43* 

44° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

/ 

0 

.64279 

.76604 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934 

60 

1 

.64301 

.76586 

.65628 

.75452 

.66935 

.74295 

.68221 

.73116 

.69487 

.71914 

59 

2 

.64323 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69508 

.71894 

58 

3 

.64346 

.76548 

.65672 

.75414 

.66978 

.74256 

.68264 

.73076 

.69529 

.71873 

57 

4 

.64368 

.76530 

.65694 

.75395 

.66999 

.74237 

.68285 

.73056 

.69549 

.71853 

56 

5 

.64390 

.76511 

.65716 

.75375 

.67021 

.74217 

.68306 

.73036 

.69570 

.71833 

55 

6 

.64412 

.76492 

.65738 

.75356 

.67043 

.74198 

.68327 

.73016 

.69591 

.71813 

54 

7 

.64435 

.76473 

.65759 

.75337 

.67064 

.74178 

.68349 

.72996 

.69612 

.71792 

53 

8 

.64457 

.76455 

.65781 

.75318 

.67086 

.74159 

.68370 

.72976 

.69633 

.71772 

53 

9 

.64479 

.76436 

.65803 

.75299 

.67107 

.74139 

.68391 

.72957 

.69654 

.71752!  51 

10 

.'64501 

.76417 

.65825 

.75280 

.67129 

.74120 

.68412 

.72937 

.69675 

.71732 

50 

11 

.64524 

.76398 

.65847 

.75261 

.67151 

.74100 

.68434 

.72917 

.69696 

.71711 

49 

12 

.64546 

.76380 

.65869 

.75241 

.67172 

.74080 

.68455 

.72897 

.69717 

.71691  48 

13 

.64568 

.76361 

.65891 

.75222 

.67194 

.74061 

.68476 

.72877 

.69737 

.71671:  47 

14 

.64590 

.76342 

.65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.69758 

.71650'  46 

15 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.72837 

.69779 

.71630  45 

16 

.64635 

.76304 

.65956 

.75165 

.67258 

.74002 

.68539 

.72817 

.69800 

.71610!  44 

17 

.64657 

.76286 

.65978 

.75146 

.67280 

.73983 

.68561 

.72797 

.69821 

.71590;  43 

18 

.64679 

.76267 

.66000 

.75126 

.67301 

.73963 

.68582 

.72777 

.69842 

.71569  42 

19 

.64701 

.76248 

.66022 

.75107 

.67323 

.73944 

.68603 

.72757 

.69862 

.71549!  41 

20 

.64723 

.76229 

.66044 

.75088 

.67344 

.73924 

.68624 

.72737 

.69883 

.71529 

40 

21 

.64746 

.76210 

.66066 

75069 

.67366 

.73904 

.68645 

.72717 

.69904 

.71508 

39 

22 

.64768 

.76192 

.66088 

75050 

.67387 

.73885 

.68666 

.72697 

.69925 

.71488 

38 

23 

.64790 

.76173| 

.68109 

75030 

.67409 

.73865 

1  .68688 

.72677 

.69946 

.71468  37 

24 

.64812 

.  76154  ! 

.66131 

75011 

.67430 

.73846 

.68709 

.72657, 

.69966  .71447 

36 

25 

.64834 

.76135 

.66153 

74992 

.67452 

.73826 

.68730 

.72637; 

.69987 

.71427 

35 

26 

.64856 

.76116 

.66175 

74973 

.67473 

.73806 

.68751 

.72617i 

.70008 

.71407  34 

27 

.64878 

.76097! 

.66197 

74953 

.67495 

.73787 

.68772 

.72597 

.70029 

.71386  33 

28 

.64901 

76078 

.66218 

74934 

.67516 

.73767 

.68793 

.  72577 

.70049 

.71366  32 

29 

.64923 

76059 

.66240 

74915 

.67538 

.73747 

.68814 

.72557 

.70070 

.71345  31 

30 

.64945 

76041 

.66262 

74896 

.67559 

.73728 

.68835 

.72537 

.70091 

.71325 

30 

31 

.64967 

.76022 

.66284 

74876 

.67580 

.73708 

.68857 

.72517 

.70112 

.71305 

29 

32 

.64989 

76003 

.66306 

74857 

.67602 

.73688 

.68878 

.72497 

70132 

.71284 

28 

33 

.65011 

75984; 

.66327 

74838 

.67623 

.73669 

.68899 

.72477 

.70153 

.71264 

27 

34 

.65033 

75965 

.66349 

74818 

.67645 

.73649 

.68920 

.72457 

.70174 

.71243 

26 

35 

.65055 

75946 

.66371 

74799 

.67666 

.73629 

.68941 

.72437 

.70195 

.71223 

35 

36 

.65077 

75927  i 

.66393 

74780 

.67688 

.73610 

.68962 

.72417 

.70215 

.71203 

24 

37 

.65100 

.75908 

.66414 

74760 

.67709 

.73590 

.68983 

.72397 

.70236 

.71182 

23 

38 

.65122 

.75889 

.66436 

74741 

.67730 

73570 

.69004 

.72377 

.70257 

.71162 

22 

39 

.65144 

75870 

.66458 

74722 

.67752 

73551 

.69025 

.72357 

.70277 

.71141 

21 

40 

.65166 

.75851 

.66480 

74703 

.67773 

73531 

.69046 

.72337 

.70298 

.71121 

20 

41 

.65188 

.75832 

.66501 

74683 

.67795 

.73511 

.69067 

.72317 

.70319 

.71100 

19 

42 

.65210 

.75813 

.66523 

74664 

.67816 

.73491 

.69088 

.72297 

.70339 

.71080 

18 

43 

.65232 

.75794 

.66545 

74644 

.67837 

.73472 

.69109 

.72277 

.70360 

.71059 

17 

44 

.65254 

.75775 

.66566 

74625 

.67859 

.73452 

.69130 

.72257 

.70381 

.71039  16 

45 

.65276 

.75756 

.66588 

74606 

.67880 

.73432 

.69151 

.72236 

.70401 

.710191  15 

46 

.65298 

.75738 

.66610 

74586 

.67901 

.73413 

.69172 

.72216 

.70422 

.70998!  14 

47 

.65320 

75719 

.66632 

74567 

.67923 

.73393 

.69193 

.72196 

.70443  .70978!  13 

48 

.65342  .75700 

.66653 

74548 

.67944 

.73373 

.69214 

.72176 

.70463  .70957  12 

49 

.65364 

.75680 

.66675 

74528 

.67965 

.73353 

.69235 

.72156 

.704841.70937  11 

50 

.65386 

.75661 

.66697 

74509 

.67987 

.73333 

.69256 

.72136 

.70505 

.70916 

10 

51 

.65408 

.75642 

.66718 

74489 

.68008 

.73314 

.69277 

.72116 

.70525 

.70896 

9 

52 

.65430 

.75623 

.66740 

74470! 

.68029 

.73294 

.69298 

.72095 

.70546  '.70875  8 

53 

.65452 

.75604 

.66762 

744511 

.68051 

.73274 

.69319 

.72075 

.705671.70855  7 

54 

.65474 

.75585! 

.66783 

7443H 

.68072 

.73254 

.69340 

.72055 

.70587  .70834  6 

55 

.65496 

.75566 

.66805 

.74412 

.68093 

.73234 

.69361 

.72035 

.70608  .70813!  5 

56 

.65518 

.75547 

.66827 

.74392 

.68115 

.73215 

.69382 

.72015 

.706281.70793  4 

57 

.65540 

.75528 

.66848  .74373 

.68136 

.73195 

.69403 

.71995 

.70649 

.70772  3 

58 

.65562 

.75509! 

.668701.74353 

.68157 

.73175 

.69424 

.71974 

.70670 

.70752:  2 

59 

.65584 

.75490! 

.  66891  !.74a34 

.68179 

.73155 

.69445 

.71954 

.70690  .70731  1 

60 

.65606 

.75471 

.66913  .74314 

.68200 

.73135 

.69466 

.71934 

.70711  .70711  0 

/ 

Cosin 

Sine 

Cosin  Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  |  Sine 

/ 

49° 

48° 

47° 

46° 

45- 

TABLE  II. 

NATURAL  TANGENTS  AND  COTANGENTS 


TO 


FIVE  DECIMAL  PLACES. 


TABLE  II. — TANGENTS   AND    COTANGENTS. 


0' 

1° 

2° 

3° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

60 

1 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.05270 

18.9755 

59 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.1664 

.05299 

18.8711 

58 

3 

.00087 

1145.92 

.01833 

54.5613 

.03579 

27.9372 

.05328 

18.7678 

57 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656 

56 

5 

.00145 

687.549 

.01891 

52.8821 

.03638 

27.4899 

.05387 

18.5645 

55 

6 

.00175 

572.957 

.01920 

52.0807 

.03667 

27.2715 

.05416 

18.4645 

54 

7 

.00204 

491.106 

.01949 

51.3032 

.03696 

27.0566 

.05445 

18.3655 

53 

8 

.00233 

429.718 

.01978 

50.5485 

.03725 

26.8450 

.05474 

18.2677 

52 

9 

.00262 

381.971 

.02007 

49.8157 

.03754 

26.6367 

.05503 

18.1708 

51 

10 

.00291 

&4S.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

50 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

.05562 

17.9802 

49 

12 

.00349 

286.478 

.02095 

47.7395 

.03842 

26.0307 

.05591 

17.8863 

48 

13 

.00378 

264.441 

.02124 

47.0853 

.03871 

25.8348 

.05620 

17.7934 

47 

14 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6418 

.05649 

17.7015 

46 

15 

.00436 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.6106 

45 

16 

.00405 

214.858 

.02211 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

17 

.00495 

202.219 

.02240 

44.G386 

.03987 

25.0798 

.05737 

17.4314 

43 

18 

.00524 

190.984 

.02269 

44.0061 

.04016 

24.8978 

.05766 

17.3432 

42 

19 

.00553 

180.932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558 

41 

20 

.00582 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.05854 

17.0837 

39 

22 

.00640 

156.259 

.02386 

41.9158 

.04133 

24.1957 

.05883 

16.9990 

38 

23 

.00669 

149.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.9150 

37 

24 

.00698 

143.237 

.02444 

40,9174 

.04191 

23.8593 

.05941 

16.8319 

36 

25 

.00727 

137.507 

.02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

35 

26 

.00756 

132.219 

.02502 

39.9655 

.04250 

23.5321 

.05999 

16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

33 

28 

.00815 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.06058 

16.5075 

32 

29 

.00844 

118.540 

.02589 

38.61:7 

.04337 

23.0577 

.06087 

16.4283 

31 

30 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

30 

31 

.00902 

110.892 

.02648 

37.7686 

.04395 

22.7519 

.06145 

16.2722 

29 

32 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.06175 

16.1952 

28 

33 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

27 

34 

.00989 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

26 

35 

.01018 

98.2179 

.02764 

35.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

36 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

37 

.01076 

92.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

38 

.01105 

90.4633 

.02851 

35.0695 

.04599 

21.7426 

.06350 

15.7483 

22 

39 

.01135 

88.1436 

.02881 

34.7151 

.04628 

21.6056 

.06379 

15.6762 

21 

40 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6048 

20 

41 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

19 

42 

.01222 

81.8470 

.02968 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

18 

43 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

.06496 

15.3943 

17 

44 

.01280 

78.1263 

.03026 

33.0452 

.04774 

20.9460 

.06525 

15.3254 

10 

45 

.01309 

76.3900 

.03055 

32.7303 

.04803 

20.8188 

.06554 

15.2571 

15 

46 

.ciass 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

i  .06584 

15.1893 

14 

47 

.01367 

73.1390 

.03114 

32.1181 

.04862 

20.5691 

.06613 

15.1222 

13 

48 

.01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0557 

12 

49 

.01425 

70.1533 

.03172 

31.5284 

.04920 

20.3253 

.06671 

14.9898 

11 

50 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

1  .06700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14.8596 

9 

52 

.01513 

66.1055 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14.7954 

8 

53 

.01542 

64.8580 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14.7317 

7 

54 

.01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

6 

55 

.01600 

62.4992 

.03346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

5 

56 

.01629 

61.3829 

.03376 

29.6245 

.05124 

19.5156 

.06876 

14.5438 

4 

57 

.01658 

60.3058 

.03405 

29.3711 

.05153 

19.4051 

.06905 

14.4823 

3 

58 

.01687 

59.2659 

.03434 

29.1220 

.05182 

19.2959 

.06934 

14.4212 

2 

59 

.01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.06963 

14.3607 

1 

60 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

.06993 

14.3007 

0 

/ 

Cotang 

Tang  ' 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

89° 

88° 

87« 

86° 

TABLE   II.      TANGENTS   AND   COTANGENTS. 


4° 

5°            1 

6°                         7°            | 

1 

Tang 

Cotang 

Tang 

Cotang 

Tang     Cotang 

Tang 

Cotang 

0 

.06993 

14.3007 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

60 

1 

.07022 

14.2411 

.08778 

11.3919 

.10540 

9.48781 

.12308 

8.12481 

59 

2 

.07051 

14.1821 

.08807 

11.3540 

.10569 

9.46141 

.12338 

8.10536 

58 

3 

.07080 

14.1235 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.08600 

57 

4 

.07110 

14.0655 

.08866 

11.2789 

.10628 

9.40904 

.12397 

8.06674 

56 

5 

.07139 

14.0079 

.08895 

11.2417 

.10657 

9.38307 

.12426 

8.04756 

55 

6 

.07168 

13.9507 

.08925 

11.2048 

.10687 

9.35724 

.12456 

8.02848 

54 

7 

.07197 

13.8940 

.08954 

11.1681 

.10716 

9.33155 

.12485 

8.00948 

53 

8 

.07227 

13.8378 

.08983 

11.1316 

.10746 

9.30599 

.12515 

7.99058 

52 

9 

.07256 

13.7821 

.09013 

11.0954 

.10775 

9.28058 

.12544 

7.97176 

51 

10 

.07285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

7.95302 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016 

.12603 

7.93438 

49 

12 

.07344 

13.6174 

.09101 

10.9882 

.10863 

9.20516 

.12633 

7.91582 

48 

13 

.07373 

13.5634 

.09130 

10.9529 

.10893 

9.18028 

.12662 

7.89734 

47 

14 

.07402 

13.5098 

.09159 

10.9178 

.10922 

9.15554 

.12692 

7.87895 

46 

15 

.07431 

13.4566 

.09189 

10.8829 

.10952 

9.13093 

.12722 

7.86064 

45 

16 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

7.84242 

44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

7.82428 

43 

18 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9.05789 

.12810 

7.80622 

42 

19 

.07548 

13.2480 

.09306 

10.7457 

.11070 

9.03379 

.12840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.00983 

.12869 

7.77035 

40 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

7.75254 

39 

22 

.07636 

13.0958 

.09394 

10.6450 

.11158 

8.96227 

.12929 

7.73480 

38 

23 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

7.71715 

37 

24 

.07695 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

7.69957 

36 

25 

.07724 

12.9469 

.09482 

10.5462 

.11246 

8.89185 

.13017 

7.68208 

35 

26 

.07753 

12.8981 

.09511 

10.5136 

.11276 

8.86862 

.13047 

7.66466 

34 

27 

.07782 

12.8496 

.09541 

10.4813 

.11305 

8.84551 

.13076 

7.64732 

33 

28 

.07812 

12.8014 

.09570 

10.4491 

.11335 

8.82252 

.13106 

7.63005 

32 

29 

.07841 

12.7536 

.09600 

10.4172 

.11364 

8.79964 

.13136 

7.61287 

31 

30 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

.13165 

7.59575 

30 

31 

.07899 

12.6591 

.09658 

10.3538 

.11423 

8.75425 

.13195 

7.57872 

29 

32 

.07929 

12.6124 

.09688 

10.3224 

.11452 

8.73172 

.13224 

7.56176 

28 

33 

.07958 

12.5660 

.09717 

10.2913 

.11482 

8.70931 

.13254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602 

.11511 

8.68701 

.13284 

7.52806 

26 

35 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.13313 

7.51132 

25 

36 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.13343 

7.49465 

24 

37 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.62078 

.13372 

7.47806 

23 

38 

,08104 

12.3390 

.09864 

10.1381 

.11639 

8.59893 

.13402 

7.46154 

22 

39 

.08134 

12.2946 

.09893 

10.1080 

.11669 

8.57718 

.13432 

7.44509 

21 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688 

8.55555 

.13461 

7.42871 

20 

41 

.08192 

12.2067 

.09952 

10.0483 

.11718 

F.  53402 

.13491 

7.41240 

19 

42 

.08221 

12.1632 

.09981 

10.0187 

.11747 

8.51259 

.13521 

7.39616 

18 

43 

.08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

17 

44 

.08280 

12.0772 

.10040 

9.96007 

.11806 

8.47007 

.13580 

7.36389 

16 

45 

.08309 

12.0346 

.10069 

9.93101 

.11836 

8.44896 

.13609 

7.34786 

15 

46 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

7.33190 

14 

47 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600 

13 

48 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

7.30018 

12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442 

11 

50 

.08456 

11.8262 

.10216 

9.78817 

.11983 

8.34496 

.13758 

7.26873 

10 

51 

.08485 

11.7853 

.10246 

8.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7'448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

7.23754 

8 

53 

.08544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

7.22204 

7 

54 

.08573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

7.20661 

6 

55 

.08602 

11.6248 

.10363 

9.64935 

.12131 

8.24345 

.13906 

7.19125 

5 

56 

.08632 

11.5853 

.10393 

9.62205 

.12160 

8.22344 

.13935 

7.17594 

4 

57 

.08661 

11.5461 

.10422 

9.59490 

.12190 

8.20352 

.13965 

7.16071 

3 

58 

.08690 

11  5072 

.10452 

9.56791 

.12219 

8.18370 

.13995 

7.14553 

2 

59 

.08720 

11.4685 

.10481 

9.54106 

.12249 

8.16398 

.14024 

7.13042 

1 

60 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

.14054 

7.11537 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

85° 

84° 

83° 

82° 

142 


TABLE   II.      TANGENTS   AND   COTANGENTS. 


8* 

1           9* 

1           10° 

11° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.14054 

7.11537 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

60 

1 

.14084 

7.10038 

.15868 

6.30189 

.17663 

5.66165 

.19468 

5.13658 

59 

2 

.14113 

7.08546 

.15898 

6.29007 

.17693 

5.65205 

.19498 

5.12862 

58 

3 

.14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19529 

5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.19559 

5.11279 

56 

5 

.14202 

7.04105 

.15988 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

55 

6 

.14232 

7:02637 

.16017 

6.24321 

.17813 

5.61397 

.19619 

5.09704 

54 

7 

.14262 

6.91174 

.16047 

6.23160 

.17843 

5.60452 

.19649 

5.08921 

53 

8 

.14291 

6.99718 

.16077 

6.22003 

.17873 

5.59511 

.19680 

5.08139 

52 

9 

.14321 

6.98268 

.16107 

6.20851 

.17903 

5.58573 

!   .19710 

5.07360 

51 

10 

.14351 

6.96823 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.06584 

50 

11 

.14381 

6.95385 

.16167 

6.18559 

.17963 

5.56706 

.19770 

5.05809 

49 

12 

.14410 

6.93952 

.16196 

6.17419 

.17993 

5.55777 

.19801 

5.05037 

48 

13 

.14440 

6.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.04267 

47 

14 

.14470 

6.91104 

.16256 

6.15151 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

.14499 

6.89688 

.16286 

6.14023 

.18083 

5.53007 

.19891 

5.02734 

45 

16 

.14529 

6.88278 

.16316 

6.12899 

.18113 

5.52090 

.19921 

5.01971 

44 

17 

.14559 

6.86874 

.16346 

6.11779 

.18143 

5.51176 

.19952 

5.01210 

43 

18 

.14588 

6.85475 

.16376 

6.10664 

.18173 

5.50264 

.19982 

5.00451 

42 

19 

.14618 

6.84082 

.16405 

6.09552 

.18203 

5.49356 

.20012 

4.99695 

41 

20 

.14648 

6.82694 

.16435 

6.08444 

.18233 

5.48451 

.20042 

4.98940 

40 

21 

.14678 

6.81312 

.16465 

6.07340 

.18263 

5.47548 

.20073 

4.98188 

39 

22 

.14707 

6.79936 

.16495 

6.06240 

.18293 

5.46648 

.20103 

4.97438 

38 

23 

.14737 

6.78564 

.16525 

6.05143 

.18323 

5.45751 

.20133 

4.96690 

37 

24 

.14767 

6.77199 

.16555 

6.04051 

.18353 

5.44857 

.20164 

4.95945 

36 

25 

.14796 

6.75838 

.16585 

6.02962 

.18384 

5.43966 

.20194 

4.95201 

35 

26 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

.14856 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

33 

28 

.14886 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

.20285 

4.92984 

32 

29 

.14915 

6.70450 

.16704 

5.98646 

.18504 

5.40429 

.20315 

4.92249 

31 

30 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20346 

4.91516 

30 

31 

.14975 

6.67787 

.16764 

5.96510 

.18564 

5.38677 

.20376 

4.90785 

29 

32 

.15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

.20406 

4.90056 

28 

33 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

27 

34 

.15064 

6.63831 

.16854 

5.93335 

.18654 

5.36070 

.20466 

4.88605 

26 

35 

.15094 

6.62523 

.16884 

5.92283 

.18684 

5.35206 

.20497 

4.87882 

25 

36 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162 

24 

37 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444 

23 

38 

.15183 

6.58627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727 

22 

39 

.15213 

6.57339 

.17004 

5.88114 

.18805 

5.31778 

.20618 

4.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

4.84300 

20 

41 

.15272 

6.54777 

.17063 

5.86051 

.18865 

5.30080 

.20679 

4.83590 

19 

42 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82882 

18 

43 

.15332 

6.52234 

.17123 

5.84001 

.18925 

5.28393 

.20739 

4.82175 

17 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4.81471 

16 

45 

.15391 

6.49710 

.17183 

5.81966 

.18986 

5.26715 

.20800 

4.80769 

15 

46 

.15421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80068 

14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

13 

48 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

4.78673 

12 

49 

.15511 

6.44720 

.17303 

5.77936 

.19106 

5.23391 

.20921 

4.77978 

11 

50 

.15540 

6.43484 

.17333 

5.76937 

.19136 

5.22566 

.20952 

4.77286 

10 

51 

.15570 

6.42253 

.17363 

5.75941 

.19166 

5.21744 

.20982 

4.76595 

9 

52 

.15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21013 

4.75906 

8 

53 

.15630 

6.39804 

.17423 

5.73960 

.19227 

5.20107 

.21043 

4.75219 

7 

54 

.15660 

6.38587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

4.74534 

6 

55 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851 

5 

56 

.15719 

6.36165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170 

4 

57 

.15749 

6.34961 

.17543 

5.70037 

.19347 

5.16863 

.21164 

4.72490 

2 

58 

.15779 

6.33761 

.17573 

5.69064 

.19378 

5.16058 

.21195 

4.71813 

S 

59 

.15809 

6.32566 

.17603 

5.68094  ! 

.19408 

5.15256 

.21225 

4.71137 

1 

60 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

.21256 

4.70463 

0 

t 

Cotaug 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

81° 

80° 

79° 

78° 

TABLE   II.      TANGENTS 


COTANGENTS. 


12° 

13° 

14° 

15° 

t 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

f 

"o 

.21256 

4.70463 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

60 

1 

.21286 

4.69791 

.23117 

4.32573 

.24964 

4.00582 

.26826 

3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001 

.24995 

4.00086 

.26857 

3.72338 

58 

3 

.21347 

4.68452 

.23179 

4.31430 

.25026 

3.99592 

.26888 

3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30860 

.25056 

3.99099 

.26920 

3.71476 

56 

5 

.21408 

4.67121 

.23240 

4.30291 

.25087 

3.98607 

.26951 

3.71046 

55 

6 

.21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

52 

9 

.215S9 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

51 

10 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

12 

.21621 

4.62518 

.23455 

4.26352 

.25304 

'3.95196 

.27169 

3.68061 

48 

13 

.21651 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.67638 

47 

14 

.21682 

4.61219 

.23516 

4.25239 

.25366 

3.94232 

.27232 

3.67217 

46 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.27263 

3.66796 

45 

16 

.21743 

4.59927 

.23578 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

44 

17 

.21773 

4.59283 

.23608 

4.23580 

.25459 

3.92793 

.27326 

3.65957 

43 

18 

.21804 

4.58641 

.23639 

4.23030 

.25490 

3.92316 

.27357 

3.65538 

42 

19 

.21834 

4.58001 

.23670 

4.22481 

.25521 

3.91839 

.27388 

3.65121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.64705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

22 

.21925 

4.56091 

.23762 

4.20842 

.25614 

3.90417 

.27482 

3.63874 

38 

23 

.21956 

4.55458 

.23793 

4.20298 

.25645 

3.89945 

.27513 

3.63461 

37 

24 

.21986 

4.54826 

.23823 

4.19756 

.25676 

3.89474 

.27545 

3.63048 

36 

25 

.82017 

4.54196 

.23854 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

35 

26 

.22047 

4.53568 

.23885 

4.18675 

.25738 

3.88536 

.27607 

3.62224 

34 

27 

.22078 

4.52941 

.23916 

4.18137 

.25769 

3.88068 

.27638 

3.61814 

33 

28 

.22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27670 

3.61405 

32 

,29 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.27701 

3.60996 

31 

30 

.22169 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732 

3  60588 

30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.60181 

29 

32 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745 

.27795 

3.59775 

28 

33 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

34 

.22292 

4.48600 

.24131 

4.14405 

.25986 

3.&4S24 

.27858 

3.58966 

26 

35 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

36 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160 

24 

37 

.22383 

4.46764 

.24223 

4.12825 

.26079 

3.83449 

.27952 

3.57758 

23 

38 

.22414 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57357 

22 

39 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.34347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

42 

.22536 

4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.55761 

18 

43 

.22567 

4.43134 

.24408 

4.09699 

.26266 

3.80726 

.28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.54968 

16 

45 

.22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

46 

.22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.54179 

14 

47 

.22689 

4.40745 

.24532 

4.07639 

.26390 

3.78931 

.28266 

3.53785 

13 

48 

.22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.28297 

3.53393 

12 

49 

.22750 

4.39560 

.24593 

4.06616 

.26452 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

52 

.22842 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

.28423 

3.51829 

8 

53 

.22872 

4.37207 

.24717 

4.04586 

.26577 

3.76268 

.28454 

3.51441 

7 

54 

.22903 

4.36623 

.24747 

4.04081 

.26608 

3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.36040 

.24778 

4.03578 

.26639 

3.75388 

.28517 

3.50666 

5 

56 

.22964 

4.35459 

.24809 

4.03076 

.26670 

3.74950 

.28549 

3.50279 

4 

57 

.22995 

4.34879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.33723 

.24902 

4.01576 

.26764 

3.73640 

.28643 

3.49125 

1 

60 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

.28675 

3.48741 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

77° 

76° 

75° 

74° 

144 


TABLE   II.       TANGENTS   AND   COTANGENTS. 


16° 

17° 

18° 

19° 

Tang 

Cotang 

Tang 

Cotang 

Tang   |  Cotang 

Tang 

Cotang 

0 

.28675 

3.48741 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

60 

1 

.28706 

3.48359 

.30605 

3.26745 

.32524 

3.07464 

.34465 

2.90147 

59 

2 

.28738 

3.47977 

.30637 

3.26406 

.32556 

3.07160 

.34498 

2.89873 

58 

3 

.28769 

3.47596 

.30669 

3.26067 

.32588 

3.06857 

.34530 

2.89600 

57 

4 

.28800 

3.47216 

.30700 

3.25729 

.32621 

3.06554 

.34563 

2.89327 

56 

5 

.28832 

3.46837 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.89055 

55 

6 

.28864 

3.46458 

.30764 

3.25055 

.32685 

3.05950 

.34628 

2.88783 

54 

7 

.28895 

3.46080 

.30796 

3.24719 

.32717 

3.05649 

.34661 

2.88511 

53 

8 

.28927 

3.45703 

.30828 

3.24383 

.32749 

3.05349 

.34693 

2.88240 

52 

9 

.28958 

3.45327 

.30860 

3.24049 

.32782 

3.05049 

.34726 

2.87970 

51 

10 

.28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2  87700 

50 

11 

.29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.87430 

49 

12 

.29053 

3.44202 

.30955 

3.23048 

.32878 

3.04152 

.34824 

2.87161 

48 

13 

.29084 

3.43829 

.30987 

3.22715 

.32911 

3.03854 

.34856 

2.86892 

47 

14 

.29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556 

.34889 

2.86624 

46 

15 

.29147 

3.43084 

.31051 

3.22053 

.32975 

3.03260 

.34922 

2.86356 

45 

16 

.29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089 

44 

17 

.29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.85822 

43 

13 

.29242 

3.41973 

.31147 

3.21063 

.33072 

3.02372 

.35020 

2.85555 

42 

19 

.29274 

3.41604 

.31178 

3.20734 

.33104 

3.02077 

.35052 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.33136 

3.01783 

.35085 

2.85023 

40 

21 

.29337 

3.40869 

.31242 

3.20079 

.33169 

3.01489 

.35118 

2.84758 

39 

22 

.29368 

3.40502 

.31274 

3.19752 

.33201 

3.01196 

.35150 

2.84494 

38 

23 

.29400 

3.40136 

.31306 

3.19426 

.33233 

3.00903 

.35183 

2.84229 

37 

24 

.29432 

3.39771 

.31338 

3.19100 

.33266 

3.00611 

.35216 

2.83965 

36 

25 

.29463 

3.39406 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702 

35 

26 

.29495 

3.39042 

.31402 

3.18451 

.33330 

3.00028 

.35281 

2.83439 

34 

27 

.29526 

3.38679 

.31434 

3.18127 

.33363 

2.99738 

.35314 

2.83176 

33 

28 

.29558 

3.38317 

.31466 

3.17804 

.33395 

2.99447 

.35346 

2.82914 

32 

29 

.29590 

3.37955 

.31498 

3.17481 

.33427 

2.99158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

31 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

32 

.29685 

3.36875 

.31594 

3.16517 

.33524 

2.98292 

.35477 

2.81870 

28 

33 

.29716 

3.36516 

.31626 

3.16197 

.33557 

2.98004 

.35510 

2.81610 

27 

34 

.29748 

3.36158 

.31658 

3.15877 

.33589 

2.97717 

.35543 

2.81350 

26 

35 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

.35576 

2.81091 

25 

36 

.29811 

3.35443 

.31722 

3.15240 

-  .33654 

2.97144 

.35608 

2.80833 

24 

37 

.29843 

3.35087 

.31754 

3.14922 

.33686 

2.96858 

.35641 

2.80574 

2?. 

38 

.29875 

3.34732 

.31786 

3.14605 

.33718 

2.96573 

.35674 

2.80316 

og 

39 

.29906 

3.34377 

.31818 

3.14288 

.33751 

2.96288 

.35707 

2.80059 

21 

40 

.29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

20 

41 

.29970 

3.33670 

.31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

42 

.30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.35805 

2.79289 

18 

43 

.30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155 

.35838 

2.79033 

17 

44 

.80065 

3.32614 

.31978 

3.12713 

.33913 

2.94872 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400 

.33945 

2.94591 

.35904 

2.78523 

15 

46 

.30128 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47 

.30160 

3.31565 

.32074 

3.11775 

.34010 

2.94028 

.35969 

2.78014 

13 

48 

.30192 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

12 

49     .30224 

3.30868 

.32139 

3.11153 

.34075 

2.93468 

.36035 

2.77507 

11 

50 

.30255 

3.30521 

.32171 

3.10843 

.34108 

2.93189 

.36068 

2.77254 

10 

51 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36134 

2.76750 

8 

53 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

7 

54 

.30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.28795 

.32331 

3.09298 

.34270 

2.91799  i 

.36232 

2.75996 

5 

56 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523  1 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.34335 

2.91246  | 

.36298 

2.75496 

3 

58 

.30509 

3.27767 

.32428 

3.08379 

.34368 

2.90971  I 

.36331 

2.75246 

2 

59 

.30541 

3.27426 

.32460 

3.08073 

.34400 

2.90696 

.36364 

2.74997 

1 

GO 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2^90421^ 

.36397 

2.74748 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

~$ang    ] 

Cotang 

Tang 

/ 

73° 

72° 

71°           1 

70° 

TABLE   II.      TANGENTS   AND   COTANGENTS. 


20° 

21° 

22° 

23° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.36397 

2.74748 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

60 

1 

.38430 

2.74499 

.38420 

2.60283 

.40436 

2.47302 

.42482 

2.35395 

59 

2 

.36463 

2.74251 

.38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

58 

3 

.36496 

2.74004 

.38487 

2.59831 

.40504 

2.46888 

.42551 

2.35015 

57 

4 

.36529 

2.73756 

.38520 

2.59606 

.40538 

2.46682 

.42585 

2.34825 

56 

5 

.36562 

2.73509 

.38553 

2.59381 

.40572 

2.46476 

.42619 

2.34636 

55 

6 

.36595 

2.73263 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

7 

.36628 

2.73017 

.38620 

2.58932 

.40640 

2.46065 

.42688 

2.34258 

53 

8 

.36661 

2.72771 

.38654 

2.58708 

.40674 

2.45860 

.42722 

2.34069 

52 

9 

.36694 

2.7252G 

.38687 

2.58484 

.40707 

2.45655 

.42757 

2.33881 

51 

10 

.36727 

2.72281 

.38721 

2.58261 

.40741 

2.45451 

.42791 

2.33693 

50 

11 

.36760 

8.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33505 

49 

12 

.36793 

2.71792 

1   .38787 

2.57815 

.40809 

2.45043 

.42860 

2.33317 

48 

13 

.36826 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

14 

.36859 

2.71305 

.38854 

2.57371 

.40877 

2.44636 

.42929 

2.32943 

46 

15 

.36892 

2.71062 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756 

45 

16 

.36925 

2.70819 

.38921 

2.50928 

.40945 

2.44230 

.42998 

2.32570 

44 

17 

.36958 

2.70577 

.38955 

2.56707 

.40979 

2.44027 

.43032 

2.32383 

43 

18 

,36991 

2.70335 

.38988 

2.56487 

.41013 

2.43825 

.43067 

2.32197 

42 

19 

.37024 

2.70094 

.39022 

2.56266 

.41047 

2.43623 

.43101 

2.32012 

41 

20 

.37057 

2.69853 

.39055 

2.56046 

.41081 

2.43422 

.43136 

2.31826 

40 

21 

.37090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.69371 

.39122 

2.55608 

.41149 

2.43019 

.43205 

2.31456 

38 

23 

.37157 

2.69131 

.39156 

2.55389 

.41183 

2.42819 

.43239 

2.31271 

37 

24 

.37190 

2.68892 

.39190 

2.55170 

.41217 

2.42618 

.43274 

2.31086 

36 

25 

.37223 

2.68653 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902 

35 

26    .37256 

2.68414 

.39257 

2.54734 

.41285 

2.42218 

.43343 

2.30718 

34. 

27 

.37289 

2.68175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534 

33 

28 

.37322 

2  67937 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351 

32 

29 

.37355 

2.67700 

.39357 

2.54082 

.41387 

2.41620 

.43447 

2.30167 

31 

30 

.37388 

2.67462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

.39425 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

29 

32 

.37455 

2.66989 

.39458 

2.53432 

.41490 

2.41025 

.43550 

2.29619 

28 

33 

.37488 

2.66752 

.39492 

2.53217 

.41524 

2.40827 

.43585 

2.29437 

27 

34 

.37521 

2.66516 

.39526 

2.53001 

.41558 

2.40629 

.43620 

2.29254 

26 

35 

.37554 

2.66281 

.39559 

2.52786 

.41592 

2.40432 

.43654 

2.29073 

25 

36 

.37588 

2.66046 

.39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891 

24 

37 

.37621 

2.65811 

.39626 

2.52357 

.41660 

2.40038 

.43724 

2.28710 

23 

38 

.37654 

2.65576 

.39660 

2.52142 

.41694 

2.39841 

.43758 

2.28528 

22 

39 

.37687 

2.65342 

.39694 

2.51959 

.41728 

2.39645 

.43793 

2.28348 

21 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754 

2.64875 

.39761 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2.64642 

.39795 

2.51289 

.41831 

2.39058 

.43897 

2.27806 

18 

43 

.37820 

2.64410 

.39829 

2.51076 

.41865 

2.38863 

.43932 

2.27626 

17 

44 

.37853 

2.64177 

.39862 

2.50864 

41899 

2.38668 

.43966 

2.27447 

16 

45 

.37887 

2.63945 

.39896 

2.50652 

.41933 

2.38473 

.44001 

2.27267 

15 

46 

.37920 

2.63714 

.39930 

2.50440 

.41968 

2.38279 

.44036 

2.27088 

14 

47 

.37953 

2.63483 

.39963 

2.50229 

.42002 

2.38084 

.44071 

2.26909 

13 

48 

.37986 

2.63252 

.39997 

2.50018 

.42036 

2.37891 

.44105 

2.26r30 

12 

49 

.38020 

2.63021 

.40031 

2.49807 

.42070 

2.37697 

.44140 

2.26552 

11 

50 

.38053 

2.62791 

.40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10 

51 

.38086 

2.62561 

.40098 

2.49386 

.42139 

2.37311 

.44210 

2.26196 

9 

52 

.38120 

2.62332 

.40132 

2.49177 

.42173 

2.37118 

.44244 

2.26018 

8 

53 

.38153 

2.62103 

.40166 

2.48967 

.42207 

2.36925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200 

2.48758 

.42242 

2.36733 

.44314 

2.25663 

6 

55 

.38220 

2.61646 

.40234 

2.48549 

.42276 

2.36541 

.44349 

2.25486 

5 

56 

.38253 

2.61418 

.40267 

2  48340 

.42310 

2.36349 

.44384 

2.25309 

4 

57 

.38286 

2.61190 

.40301 

2.48132 

.42345 

2.36158 

.44418 

2.25132 

3 

58 

.38320 

2.60963 

.40335 

2.47924 

.42379 

2.35967 

.44453 

2.24956 

2 

59 

.38353 

2.60736 

.40369 

2.47716 

.42413 

2.35776 

.44488 

2.24780 

1 

60 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

.44523 

2.24604 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

69° 

68'          H          67° 

66° 

TABLE   II.      TANGENTS   AND    COTANGENTS. 


2 

40 

2 

5° 

2 

6° 

2 

7° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotaug 

Tang 

Cotang 

/ 

0 

.44523 

2.24604 

.46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

60 

1 

.44558 

2.24428 

.46666 

2.14288 

.48809 

2.04879 

.50989 

1.96120 

59 

2 

.44593 

2.24252 

.46702 

2.14125 

.48845 

2.04728 

.51026 

1.95979 

58 

3 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

1.95838 

57 

4 

.44662 

2.23902 

.46772 

2.13801 

.48917 

2.04426 

.51099 

1.95698 

56 

5 

.44697 

2.23727 

.46808 

2.13639 

.48953 

2.04276 

.51136 

1.95557 

55 

6 

.44732 

2.23553 

.46843 

2.13477 

.48989 

2.04125 

.51173 

1.95417 

54 

7 

.44767 

2.23378 

.46879 

2.13316 

.49026 

2.03975 

.51209 

1.95277 

53 

8 

.44802 

2.23204 

.46914 

2.13154 

.49062 

2.03825 

.51246 

1.95137 

52 

9 

.44837 

2.23030 

.46950 

2.12993 

.49098 

2.03675 

.51283 

1.94997 

51 

10 

.44872 

2.22857 

.46985 

2.12832 

.49134 

2.03526 

.51319 

1.94858 

50 

11 

.44907 

2.22683 

.47021 

2.12671 

.49170 

2.03376 

.51356 

.94718 

49 

12 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

.94579 

48 

13 

.44977 

2.22337 

.47092 

2.12350 

.49242 

2.03078 

.51430 

.94440 

47 

14 

.45012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

.94301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

.94162 

45 

16 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

.94023 

44 

17 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

.93885 

43 

18 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.51614 

.93746 

42 

19 

.45187 

2.21304 

.47305 

2.11392 

.49459 

2.02187 

.51651 

.93608 

41 

20 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

1.93332 

39 

22 

.45292 

2.20790 

.47412 

2.10916 

.49568 

2.41743 

.51761 

1.93195 

38 

23 

.45327 

2.20619 

.47448 

2.10758 

.49604 

2.01596 

.51798 

1.93057 

37 

24 

.45362 

2.20449 

.47483 

2.10600 

.49640 

2.01449 

.51835 

1.92920 

36 

25 

.45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51872 

1.92782 

35 

26 

.45432 

2.20108 

.47555 

2.10284 

.49713 

2.01155 

.51909 

.92645 

34 

27 

.45467 

2.19938 

.47590 

2.10126 

.49749 

2.01008 

.51946 

.92508 

33 

28 

.45502 

2.19769 

.47626 

2.09969 

.49786 

2.00862 

.51983 

.92371 

32 

29 

.45538 

2.19599 

.47662 

2.09811 

.49822 

2.00715 

.52020 

.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

.92098 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49894 

2.00423 

.52094 

.91962 

29 

32 

.45643 

2.19092 

.47769 

2.09341 

.49931 

2.00277 

.52131 

.91826 

28 

33 

.45678 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

.52168 

.91690 

27 

34 

.45713 

2.18755 

.47840 

2.09028 

.50004 

1.99986 

.52205 

.91554 

26 

35 

.45748 

2.18587 

.47876 

2.08872 

.50040 

1.99841 

.52242 

.91418 

25 

36 

.45784 

2.18419 

.47912 

2.08716 

.50076 

1.99695 

.52279 

.91282 

24 

37 

.45819 

2.18251 

.47948 

2.08560 

.50113 

1.99550 

.52316 

.91147 

23 

38 

.45854 

2.18084 

.47984 

2.08405 

.50149 

1.99406 

.52353 

.91012 

22 

39 

.45889 

2.17916 

.48019 

2.08250 

.50185 

1.99261 

.52390 

.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

1.99116 

.52427 

.90741 

20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

1.98972 

.52464 

.90607 

19 

42 

.45995 

2.17416 

.48127 

2.07785 

.50295 

1.98828 

.52501 

.90472 

18 

43 

.46030 

2.17249 

.48163 

2.07630 

.50331  • 

1.98684 

.52538 

.90337 

17 

44 

.46065 

2.17083 

.48198 

2.07476 

.50368 

1.98540 

.52575 

.90203 

16 

45 

.46101 

2.16917 

.48234 

2.07321 

.50404 

1.98396 

.52613 

.90069 

15 

46 

.46136 

2.16751 

.48270 

2.07167 

.50441 

1.98253 

.52650 

.89935 

14 

47 

.46171 

2.16585 

.48306 

2.07014 

.50477 

1.98110 

.52687 

.8S801 

13 

48 

.46206 

2.16420 

.48342 

2.06860 

.50514 

1.97966 

.52724 

.89667 

12 

49 

.46242 

2.16255 

.48378 

2.06706 

.50550 

1.97823 

.52761 

.89533 

11 

50 

.46277 

2.16090 

.48414 

2.06553 

.50587 

1.97681 

.52798 

.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

1.97538 

.52836 

1.8926& 

9 

52 

.46348 

2.15760 

.48486 

2.06247 

.50660 

1.97395 

.52873 

1.89133 

8 

53 

.46383 

2.15596 

.48521 

2.06094 

.50696 

1.97253 

.52910 

1.89000 

7 

54 

.46418 

2.15432 

.48557 

2.05942 

.50733 

.97111 

.52947 

1.88867 

6 

65 

.46454 

2.15268 

.48593 

2.05790 

.50769 

.96969 

.52985 

1.88734 

5 

66 

.46489 

2.15104 

.48629 

2.05637 

.50806 

.96827 

.53022 

1.88602 

4 

67 

.46525 

2.14940 

.48665 

2.05485 

.50843 

.96685 

.53059 

1.88469 

3 

68 

.46560 

2.14777 

.48701 

2.05333 

.50879 

.96544 

.53096 

1.88337 

2 

69 

.46595 

2.14614 

.48737 

2.05182 

.50916 

.96402 

.53134 

1.88205 

1 

60 

.46631 

2.14451 

.48773 

2.05030 

.50953 

..96261 

.53171 

1.88073 

0 

f 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

6 

5° 

6 

4° 

6 

3' 

6 

P           1 

TABLE   II.      TANGENTS   AND    COTANGENTS. 


2 

8° 

2 

9° 

3 

0° 

3 

1° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.53171 

1.88073 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

60 

1 

.53208 

.87941 

.55469 

1.80281 

.57774 

1.73089 

.60126 

1.66318 

59 

2 

.53246 

.87809 

.55507 

1.80158 

.57813 

1.72973 

.60165 

1.66209 

58 

3 

.53283 

.87677 

.55545 

1.80034 

.57851 

1.72857 

.60205 

1.66099 

57 

4 

.53320 

.87546 

.55583 

1.79911 

.57890 

1.72741 

.60245 

1.65990 

56 

5 

.53358 

.87415 

.55621 

1.79788 

.57929 

1.72625 

.60284 

1.65881 

55 

6 

.53395 

.87283 

.55659 

1.79665 

.57968 

1.72509 

.60324 

1.65772 

54 

7 

.53432 

.87152 

.55697 

1.79542 

.58007 

1.72393 

.60364 

1.65663 

53 

8 

.53470 

.87021 

.55736 

1.79419 

.58046 

1.72278 

.60403 

1.65554 

52 

9 

.53507 

.86891 

.55774 

1.79296 

.58085 

1.72163 

.60443 

1.65445 

51 

10 

.53545 

.86760 

.55812 

1.79174 

.58124 

1.72047 

.60483 

1.65337 

50 

11 

.53582 

.86630 

.55850 

1.79051 

.58162 

1.71932 

.60522 

1.65228 

49 

12 

.53620 

.86499 

.55888 

1.78929 

.58201 

1.71817 

.60562 

1.65120 

48 

13 

.53657 

.86369 

.55926 

1.78807 

.58240 

1.71702 

.60602 

1.65011 

47 

14 

.53694 

.86239 

.55964 

1.78685 

.58279 

1.71588 

.60642 

1.64903 

46 

15 

.53732 

.86109 

.56003 

1.78563 

.58318 

1.71473 

.60681 

1.64795 

45 

16 

.53769 

.85979 

.56041 

1.78441 

.58357 

1.71358 

.60721 

1.64687 

44 

17 

.53807 

.85850 

.56079 

1.78319 

.58396 

1.71244 

.60761 

1.64579 

43 

18 

.53844 

.85720 

.56117 

1.78198 

.58435 

1.71129 

.60801 

1.64471 

42 

19 

.53882 

.85591 

.56156 

1.78077 

.58474 

1.71015 

.60841 

1.64363 

41 

20 

.53920 

.85462 

.56194 

1.77955 

.58513 

1.70901 

.60881 

1.64256 

40 

21 

.53957 

.85333 

.56232 

.77834 

.58552 

1.70787 

.60921 

1.64148 

39 

22 

.53995 

.85204 

.56270 

.77713 

.58591 

1.70673 

.60960 

1.64041 

38 

23 

.54032 

.85075 

.56309 

.77592 

.58631 

1.70560 

.61000 

1.63934 

37 

24 

.54070 

.84946 

.56347 

.77471 

.58670 

1.70446 

.61040 

1.88826 

36 

25 

.54107 

.84818 

.56385 

.77351 

.58709 

1.70332 

.61080 

1.63719 

35 

26 

.54145 

.84689 

.56424 

.77230 

.58748 

1.70219 

.61120 

1.63612 

34 

27 

.54183 

.84561 

.56462 

.77110 

.58787 

1.70106 

.61160 

1.63505 

33 

28 

.54220 

.84433 

.56501 

.76990 

.58826 

1.69992 

.61200 

1.63398 

32 

29 

.54258 

.84305 

.56639 

.76869 

.58865 

1.69879 

.61240 

1.63292 

31 

30 

':54296 

1.84177 

.56577 

1.76749 

.58905 

1.69766 

.61280 

1.63185 

30 

31 

.54333 

1.84049 

.56616 

1.76629 

.58944 

1.69653 

.61320 

1.63079 

29 

32 

.54371 

1.83922 

.56654 

1.76510 

.58983 

1.69541 

.61360 

1.62972 

28 

33 

.54409 

1.83794 

.56693 

1.76390 

.59022 

1.69428 

.61400 

1.62866 

27 

34 

.54446 

1.83667 

.56731 

1.76271 

j   .59061 

1.69316 

.61440 

1.62760 

26 

35 

.54484 

1.83540 

.56769 

1.76151 

|   .59101 

1.69203 

.61480 

1.62654 

25 

36 

.54522 

1.83413 

.56808 

1.76032 

i   .59140 

1.69091 

.61520 

1.62548 

24 

37 

.54560 

1.83286 

.56846 

1.75913 

.59179 

1.68979 

.61561 

1.62442 

23 

38 

.54597 

1.83159 

.56885 

1.75794 

.59218 

1.68866 

.61601 

1.62336 

22 

39 

.54635 

1.83033 

.56923 

1.75675 

.59258 

1.68754 

.61641 

1.62230 

21 

40 

.54673 

1.82906 

.56962 

1.75556 

.59297 

1.68643 

.61681 

1.62125 

20 

41 

.54711 

1.82780 

.57000 

.75437 

.59336 

1.68531 

.61721 

1.62019 

19 

42 

.54748 

1.82654 

.57039 

.75319 

.59376 

1.68419 

.61761 

1.61914 

18 

43 

.54786 

1.82528 

.57078 

.75200 

.59415 

1.68308 

1   .61801 

1.61808 

17 

44 

.54824 

1.82402 

.57116 

.75082 

.59454 

1.68196 

.61842 

1.61703 

16 

46 

.54862 

1.82276 

.57155 

.74964 

.59494 

1.68085 

.61882 

1.61598 

15 

46 

.54900 

.82150 

.57193 

.74846 

.59533 

1.67974 

.61922 

1.61493 

14 

47 

.54938 

.82025 

.57232 

.74728 

.59573 

1.67863 

.61962 

1.61388 

13 

48 

.54975 

.81899 

.57271 

.74610 

.59612 

1.67752 

.62003 

1.61283 

12 

49 

.55013 

.81774 

.57309 

.74492 

.59651 

1.67641 

.62043 

1.61179 

11 

50 

.55051 

.81649 

.57348 

.74375 

.59691 

1.67530 

.62083 

1.61074 

10 

51 

.55089 

.81524 

.57386 

.74257 

.59730 

1.67419 

.62124 

1.60970 

9 

52 

.55127 

.81399 

.57425 

.74140 

.59770 

1.67309 

.62164 

1.60865 

8 

53 

.55165 

.81274 

.57464 

.74022 

.59809 

1.67198 

.62204 

1.60761 

7 

54 

.55203 

.81150 

.57503 

.73905 

.59849 

1.67088 

.62245 

1.60657 

6 

55 

.55241 

.81025 

.57541 

.73788 

.59888 

1.66978 

.62285 

1.60553 

5 

56 

.55279 

.80901 

.57580 

.73671 

.59928 

1.66867 

.62325 

1.60449 

4 

57 

.55317 

.80777 

.57619 

.73555 

.59967 

1.66757 

.62366 

1.60345 

3 

58 

.55355 

.80653 

.57657 

.73438 

.60007 

1.66647 

.62406 

1.60241 

2 

59 

.55393 

.80529 

.57696 

.73321 

.60046 

1.66538 

.62446 

1.60137 

1 

60 

.55431 

1.80405 

.57735 

.73205 

.60086 

1.66428 

.62487 

1.60033 

0 

t 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

6 

1° 

6 

0° 

5 

3° 

$! 

J° 

TABLE  II.-TANGENTS  AND  COTANGENTS. 


32°            p            33° 

34°             !            35° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang  !    Tang 

Cotang 

0    .62487 

1.60033 

.64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

0 

1     .62527 

1.59930 

.64982 

1.53888 

.67493 

1.48163 

.70064 

1.42726 

9 

2    .62568 

1.59826 

,65024 

1.53791 

.67536 

1.48070 

.70107 

1.42638 

8 

3    .62608 

1.59723 

.65065 

1.53693 

.67578 

1.47977 

.70151 

1.42550 

7 

4     .62649 

1.59620 

.65106 

1  .  53595 

.67620 

1.47885 

.70194 

1.42462 

6 

5     .62689 

1.59517 

.65148 

1.53497 

.67663 

1.47792 

.70238 

1.42374 

5 

6     .62730 

1.59414 

.65189 

1.53400 

.  67705 

1.47699 

.70281 

1.42286 

4 

7    .62770 

1.59311 

.65231 

1.53302 

.67748 

1.47607 

.70325 

1  42198 

53 

8    .62811 

1.59208 

.65272 

1.53205 

.67790 

1.47514 

.70368 

1.42110 

52 

9    .62852 

1.59105 

.65314 

1.53107 

.67832 

1.47422 

.70412 

1.42022 

ol 

10    .62892 

1.59002 

.65355 

1.53010 

.67875 

1.47330 

.70455 

1.41934 

50 

11     .629S3 

1.58900 

.65397 

1.52913 

.67917 

1.47238 

.70499 

1.41847 

49 

12    .62973 

1.58797 

.65438 

1.52816 

.67960 

1.47146 

.70542 

1.41759 

48 

13     .63014 

1.58695 

.65480 

1.52719 

.68002 

1.47053 

.70586 

1.41672 

47 

14     .63055 

1.58593 

.65521 

1.52622 

.68045 

1.46962 

.70629 

1.41584 

46 

15     .63095 

1.58490 

.65563 

1.52525 

.68088 

1.46870 

.70673 

1.41497 

45 

16     .63136 

1.58388 

.65604 

1.52429 

.68130 

1.46778 

.70717 

1.41409 

44 

17    .63177 

1.58286 

.65646 

1.52332 

.68173 

1.46686 

.70760 

1.41322 

43 

18     .63217 

1.58184 

.65688 

1.52235 

.68215 

1.46595 

.70804 

1.41235 

42 

19      63258 

1.58083 

.65729 

1.52139 

.68258 

1.46503 

.70848 

1.41148 

41 

20    .63299 

1.57981 

.65771 

1.52043  ; 

.68301 

1.464H 

.70891 

1.41061 

40 

21     .63340 

1.57879 

.65813  ' 

1.51946 

.68343 

1.46320 

.70935 

1.40974 

39 

22     .63380 

1.57778 

.65854 

1.51850 

.08386 

1.46229 

.70979 

1.40887 

38 

23     .63421 

1.57676 

.65896 

1.51754 

.68429 

1.46137 

.71023 

1.40800 

37 

24     .63462 

1.57575 

.65938 

1.51658 

.68471 

1.46046 

.71066 

1.40714 

36 

25     .63503 

1.57474 

.65980 

1.51562 

.68514 

1.45955 

.71110 

1.40627 

35 

26    .63544 

1.57372 

.66021 

1.51466 

.68557 

1.45864 

.71154 

1.40540 

*4 

27     .63584 

1.57^1 

.66063 

1.51370 

.68600 

1.4577'3 

.71198 

1.40454 

33 

28     .63625 

1.57170 

.66105 

1.51275 

.68642 

1.45682 

.71242 

1.40367 

32 

29    .63666 

1.57069 

.66147 

1.51179 

.68685 

1.45592 

.71285 

1.40281 

31 

30    .63707 

1.56969 

.66189 

1.51084 

.68728 

1.45501 

.71329 

1.40195 

30 

31     .63748 

1.56868 

.66230 

1.50988 

.68771 

1.45410 

.71373 

1.40109 

29 

32     .63789 

1.56767 

.66272 

1.50893 

.68814 

1.45320 

.71417 

1.40022 

28 

33    .63830 

1.56667 

.66314 

1.507'97 

.68857 

1.45229 

.71461 

1.39936 

27 

34    .63871 

1.56566 

.66356 

1.50702 

.68900 

1.45139 

.71505 

1.39850 

26 

35    .63912 

1.56466 

.66398 

1.50607 

.68942 

1.45049 

.71549 

1.39764 

25 

36    .63953 

1.56366 

.66440 

1.50512 

.68985 

1.44958 

.71593 

1.39679 

24 

37    .63994 

1.56265 

.66482 

1.50417 

.69028 

1.44868 

.71637 

1.39593 

23 

38     .64035 

1.56165 

.66524 

1.50322 

.69071 

1.44778 

.71681 

1.39507 

22 

39    .64076 

1.56065 

.66566 

1.50228 

.69114 

1.44688 

.71725 

1.39421 

21 

40    .64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

.71769 

1.39336 

20 

41     .64158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

.71813 

1.39250 

19 

42    .64199 

1.55766 

.66692 

1.49944 

.69243 

1.44418 

.71857 

1.39165 

18 

43     .64240 

1.55666 

.66734 

1.49849 

.69286 

1.44329 

.71901 

1.39079 

17 

44     .64281 

1.55567 

.66776 

1.49755 

.69329 

1.44239 

.71946 

1.38994 

1C 

45     .64322 

1.55467 

.66818 

1.49661 

.69372 

1.44149 

.71990 

1.38909 

15 

46     .64363 

1.55368 

.66860 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14 

47     .64404 

1.55269 

.66902 

1.49472 

.69459 

1.43970 

.72078 

1.38738 

13 

48     .64446 

1.55170 

.66944 

1.49378 

.69502 

1.43881 

.72122 

1.38653 

12 

49     .64487 

1.55071 

.66986 

1.49284 

.69545 

1.43792 

.72167 

1.385G8 

11 

50    .64528 

1.54972 

.67028 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

10 

51     .64569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.38399 

9 

52     .64610 

1.54774 

.67113 

1.49003 

.69675 

1.43525 

.72299 

1.38314 

8 

53     .64652 

1.54675 

.67155 

1.48909 

.69718 

1.43436 

.72344 

1.38229 

r< 

54     .64693 

1.54576 

I   .67197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

6 

55    .64734 

1.54478 

1   .67239 

1.48722 

.69804 

1.43258 

.72432 

1.38060 

5 

56     .64775 

1.54379 

.67282 

1.48629 

.69847 

1.43189 

.72477 

1.37976 

i 

57    .64817 

1.54281 

.67324 

1.48536 

.69891 

1.43080 

.72521 

1.37891 

3 

58    .64858 

1.54183 

.67366 

1.48442 

.69934 

1.42992 

.72565 

1.37807 

2 

59    .64899 

1.54085 

.67409 

1.48349 

.69977 

1.42903 

.72610 

1.37722 

1 

60    .64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

.72654 

1.37638 

_o 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

i  Cotang 

Tang 

t 

57° 

i           56° 

55°           II           54* 

TABLE   It        TANGENTS   AND,  COTANGENTS. 


36° 

37° 

38^ 

39° 

/ 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

~o 

.72654 

1.37638 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

60 

1 

.72699 

1.37554 

.75401 

1.32624 

.7817.1 

1.27917 

.81027 

1.23416 

59 

2 

.72743 

1.37470 

.75447 

1.32544 

.75222 

1.27841 

.81075 

1.23343 

58 

3 

.72788 

1.37386 

.75492 

1.32464 

.78269 

1.27764 

.81123 

1.23270 

57 

4 

.72832 

1.37302 

.75538 

1.32384 

.78316 

1.27688 

.81171 

1.23196 

56 

5 

.72877 

1.37218 

.75584 

1.32304 

.78363 

1.27611 

.81220 

1.23123 

55 

6 

.72921 

1.37134 

.75629 

1.32224 

.78410 

1.27535 

.81268 

1.23050 

54 

7 

.72966 

1.37050 

.75675 

1.32144 

.78457 

1.27458 

.81316 

1.22977 

53 

8 

.73010 

1.36967 

.75721 

1.32064 

.78504 

1.27382 

.81364 

1.22904 

52 

9 

.73055 

1.36883 

.75767 

1.31984 

.78551 

1.27306 

.81413 

1.22831 

51 

10 

.73100 

1.36800 

.75812 

1.31904 

.78598 

1.27230 

.81461 

1.22758 

50 

11 

.73144 

1.36716 

.75858 

1.31825 

.78645 

1.27153 

.81510 

1.22685 

49 

12 

.73189 

1.36633 

.75904 

1.31745 

.78692 

1.27077 

.81558 

1.22612 

48 

13 

.73234 

1.36549 

.75950 

1.31666 

.78739 

1.27001 

.81606 

1.22539 

47 

14 

.73278 

1.36466 

.75996 

1.  '31586 

.78786 

1.26925 

.81655 

1.22467 

40 

15 

.73323 

1.36383 

.76042 

1.31507 

.78834 

1.26849 

.81703 

1.22394 

45 

16 

.73368 

1.36300 

.76088 

1.31427 

.78881 

1.26774 

.81752 

1.22321 

44 

17 

.73413 

1.36217 

.76134 

1.31348 

.78928 

1.26698 

.81800 

1.22249 

43 

18 

.73457 

1.36134 

.76180 

1.31269 

.78975 

1.  '26622 

.81849 

1.22176 

42 

19 

.73502 

1.36051 

.76226 

1.31190 

.79022 

1.26546 

.81898 

1.22104 

41 

20 

.73547 

1.35968 

.76272 

1.31110 

.79070 

1.26471 

.81946 

1.22031 

40 

21 

.73592 

1.35885 

.76318 

1.31031 

.79117 

1.26395 

.81995 

1.21959 

39 

22 

.73637 

1.35802 

.76364 

1.30952 

.79164 

1.26319 

.82044 

1.21886 

38 

23 

.73681 

1.35719 

.76410 

1.30873 

.79212 

1.26244 

.82092 

1.21814 

37 

24 

.73726 

1.35637 

.76456 

1.30795 

.79259 

1.26169 

.82141 

1.21742 

36 

25 

.73771 

1.35554 

.76502 

1.30716 

.79306 

1.26093 

.82190 

1.21670 

35 

26 

.73816 

1.35472 

.76548 

1.30637 

.79354 

1.26018 

.82238 

1.21598 

34 

27 

.73861 

1.35389 

.76594 

1.30558 

.79401 

1.25943 

.82287 

1.21526 

33 

28 

.73906 

1.35307 

.76640 

1.30480 

.79449 

1.25867 

.82336 

1.21454 

32 

29 

.73951 

1.35224 

.76686 

1.30401 

.79496 

1.25792 

.82385 

1.21382 

31 

30 

.73996 

1.35142 

.76733 

1.30323 

.79544 

1.25717 

.82434 

1.21310 

30 

31 

.74041 

1.35060 

.76779 

1.30244 

.79591 

1.25642 

.82483 

1.21238 

29 

32 

.74086 

1.34978 

.76825 

1.30166 

.79639 

1.25567 

.82531 

1.21166 

28 

33 

.74131 

1.34896 

.76871 

1.30087 

.79686 

1.25492 

.82580 

1.21094 

27 

34 

.74176 

1.34814 

i  .76918 

1.30009 

.79734 

1.25417 

.82629 

1.21023 

26 

85 

.74221 

1.34732 

.76964 

1.29931 

.79781 

1.25343 

.82678 

1.20951 

25 

36 

.74267 

1.34650 

.77010 

1.29853 

.79829 

1.25268 

.82727 

1.20879 

24 

37 

.74312 

1.34568 

.77057 

1.29775 

.79877 

1.25193 

.82776 

1.20808 

23 

38 

.74357 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

82825 

1.20736 

22 

39 

.74402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

1.20665 

21 

40 

.74447 

1.34323 

.77196 

1.29541 

.80020 

1.24969 

.82923 

1.20593 

20 

41 

.74492 

1.34242 

.77242 

1.29463 

.80067 

1.24895 

.82972 

1.20522 

19 

42 

.74538 

1.34160 

.77289 

1.29385 

.80115 

1.24820 

.83022 

1.20451 

18 

43 

.74583 

1.34079 

.77335 

1.29307 

.80163 

1.24746 

.83071 

1.20379 

17 

44 

.74628 

1.33998 

.77382 

1.29229 

.80211 

1.24672 

.83120 

1.20308 

16 

45 

.74674 

1.33916 

.77428 

1.29152 

.80258 

1.24697 

.83169 

1.20237 

15 

46 

.74719 

1.33835 

.77475 

1.29074 

.80306 

1.24523 

.83218 

1.20166 

14 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

1.28919 

.80402 

1.24375 

.83317 

1.20024 

12 

49 

.74855 

1.33592 

.77615 

1.28842 

.80450 

1.24301 

.83366 

1.19953 

11 

50 

.74900 

1.33511 

.77661 

1.28764 

.80498 

1.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80546 

1.24153 

.83465 

1.19811 

9 

52 

.74991 

1.33349 

.77754 

1.28610 

.80594 

1.24079 

.83514 

1.19740 

8 

53 

.75037 

1.33268 

.77801 

1.28533 

.80642 

1.24005 

.83564 

1.19669 

7 

54 

.75082 

1.33187 

.77848 

1.28456 

.80690 

1.23931 

.83613 

1.19599 

6 

55 

.75128 

1.33107 

.77895 

1.28379 

.80738 

1.23858 

.83662 

1.19528 

5 

56 

.75173 

1.33026 

.77941 

1.28302 

.80786 

1.23784 

.83712 

1.19457 

4 

57 

.75219 

1.32946 

.77988 

1.28225 

.80834 

1.23710 

.83761 

1.19387 

3 

58 

.75264 

1.32865 

.78035 

1.28148 

.80882 

1.23637 

.83811 

1.19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

.80930 

1.23563 

.83860 

1.19246 

1 

60 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

.83910 

1.19175 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

53° 

52° 

51° 

50° 

TABLE   II.      TANGENTS   AND   COTANGENTS. 


40° 

41° 

42° 

43° 

Tang  |  Cotang 

Tang 

Cotang 

Tang     Cotang 

Tang 

Cotang 

0 

.83910 

1.19175 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

60 

1 

.83960 

1.19105 

.86980 

1.14969 

.90093 

1.10996 

.93306 

1.07174 

59 

2 

.84009 

1.19035 

.87031 

1.14902 

.90146 

1.10931 

.93360 

1.07112 

58 

3 

.84059 

1.18964 

.87082 

1.14834 

.90199 

1.10867 

.93415 

1.07049 

57 

4 

.84108 

1.18894 

.87133 

1.14767 

.90251 

1.10802 

.93469 

1.06987 

56 

5 

.84158 

1.18824 

.87184 

1.14699 

.90304 

1.10737 

.93524 

1.06925 

55 

6 

.84208 

1.18754 

.87236 

1.14632 

.90357 

1.10672 

.93578 

1.06862 

54 

7 

.84258 

1.18684 

.87287 

1.14565 

.90410 

1.10607 

.93633 

1.06800 

53 

8 

.84307 

1.18614 

.87338 

1.14498 

.90463 

1.10543 

.93688 

1.06738 

52 

9 

.84357 

1.18544 

.87389 

1.14430 

.90516 

1.10478 

.93742 

1.06676 

51 

110 

.84407 

1.18474 

.87441 

1.14363 

.90569 

1.10414 

.93797 

1.06613 

50 

11 

.84457 

1.18404 

.87492 

1.14296 

.90621 

1.10349 

.93852 

1.06551 

49 

12 

.84507 

1.18334 

.87543 

1.14229 

.90674 

1.10285 

.93906 

1.06489 

43 

13 

.84556 

1.18264 

.87595 

1.14162 

.90727  1  1.10220 

.93961 

1.06427 

47 

14 

.84606 

1.18194 

.87646 

1.14095 

.90781      1.10156 

.94016 

1.06365 

46 

15 

.84656 

1.18125 

.87698 

1.14028 

.90834  !  1.10091 

.94071 

1.06303 

45 

16 

.84706 

1.18055 

.87749 

1.13961 

.90887  1  1.10027 

.94125 

1.06241 

44 

17 

.84756 

1.17986 

.87801 

1.13894 

.90940     1.09963 

.94180 

1.06179 

43 

18 

.84806 

1.17916 

.87852 

1.13828 

.90993     1.09899 

.94235 

1.06117 

42 

19 

.84856 

1.17846 

.87904 

1.13761 

.91046     1.09&34 

.94290 

1.06056 

41 

20 

.84906 

1.17777 

.87955 

1.13694 

.91099 

1.0977'0 

.94345 

1.05994 

40 

21 

.84956 

1.17708 

.88007 

1.13627 

.91153 

1.09706 

.94400 

1.05932 

39 

22 

.85006 

1.17638 

.88059 

1.13561 

.91206 

1.09642 

.94455 

1.05870 

38 

23 

.85057 

1.17569 

.88110 

1.13494 

.91259 

1.09578 

.94510 

1.05809 

37 

24 

.85107 

1.17500 

.88162 

1.13428 

.91313 

1.09514 

.94565 

1.05747 

36 

25 

.85157 

1.17430 

.88214 

1.13361 

.91366 

1.09450 

.94620 

1.05685 

35 

26 

.85207 

1.17361 

.88265 

1.13295 

.91419 

1.09386 

.94676 

1.05624 

34 

27 

.85257 

1.17292 

.88317 

1.13228 

.91473 

1.09322 

.94731 

1.05562 

33 

28    .85308 

1.17223 

.88369 

1.13162 

.91526 

1.09258 

.94786 

1.05501    32 

29    .85358 

1.17154 

.88421 

1.13096 

.91580 

1.09195 

.94841 

1.05439  !31 

30    .85408 

1.17085 

.88473 

1.13029 

.91633 

1.09131 

.94896 

1.05378 

30 

31 

.85458 

1.17016 

.88524 

1.12963 

.91687 

1.09067 

.94952 

1.05317 

29 

32 

.85509 

1.16947 

.88576 

1.12897 

.91740 

1.09003 

.95007 

1.05255 

28 

33 

.85559 

1.16878 

.88628 

1.12831 

.91794 

1.08940 

.95062 

1.05194 

27 

34 

.85609 

1.16809 

.88680 

1.12765 

.91847 

1.08876 

.95118 

1.05133 

26 

35 

.85660 

1.16741 

.88732 

1.12699 

.91901     1.08813 

.95173 

1.05072 

25 

36 

.85710 

1.16672 

,   .88784 

1.12633 

.91955     1.08749 

.95229 

1.05010 

24 

37 

.85761 

1.16603 

.88836 

1.12567 

.92008  :  1.08686 

.95284 

1.04949 

23 

38 

.85811 

1.16535 

.88888 

1.12501 

.92062 

1.08622 

.95340 

1.04888 

22 

39 

.85862 

1.16466 

.88940 

1.12435 

.92116 

1.08559 

.95395 

1.04827 

21 

40 

.85912 

1.16398 

.88992 

1.12369 

.92170 

1.08496 

.95451 

1.04766 

20 

41 

.85963 

1.16329 

.89045 

1.12303 

.92224 

1.08432 

.95506 

1.04705 

19 

42 

.86014 

1.16261 

.89097 

1.12238 

.92277 

1.08369 

.95562 

1.04644 

18 

43 

.86064 

1.16192 

.89149 

1.12172 

.92331 

1.08306 

.95618 

1.04583 

17 

44 

.86115 

1.16124 

.89201 

1.12106 

.92385     1.08243 

.95673 

.04522 

Id 

45 

.86166 

1.16056 

.89253 

1.12041 

.92439  !  1.08179 

.95729 

.04461 

15 

46 

.86216 

1.15987 

.89306 

1.11975 

.92493 

1.08116 

.95785 

.04401 

14 

47 

.86267 

1.15919 

.89358 

1.11909 

.92547 

1.08053 

.95841 

.04340 

13 

48 

.86318 

1.15851 

.89410 

1.11844 

.92601 

1-07990 

.95897 

.04279 

13 

49 

.86368 

1.15783 

.89463 

1  11778 

.92655 

1.07927 

.95952 

.04218 

11 

50 

.86419 

1.15715 

.89515 

1.11713 

.92709 

1.07864 

.96008 

.04158 

10 

51 

.86470 

1.15647 

.89567 

1.11648 

.92763 

1.07801 

.96064 

.04097 

D 

52 

.86521 

1.15579 

.89620 

1.11582 

.92817 

1.07738 

.96120 

.04036 

8 

53 

.86572 

1.15511 

.89672 

1.11517 

.92872 

1.07676 

.96176 

.03976 

7 

54 

.86623 

1.15443 

.89725 

1.11452 

.92926 

1.07613 

.96232 

.03915 

6 

55 

.86674 

1.15375 

.89777 

1.11387 

.92980 

1.07550 

.96288 

.03855 

5 

56 

.86725 

1.15308 

.89830 

1.11321 

.93034 

1.07487 

.96344 

.03794 

4 

57 

.86776 

1.15240 

.89883 

1.11256 

.93088 

1.07425 

.96400 

.03734 

3 

58 

.86827 

1.15172 

.89935 

1.11191 

.93143 

1.07362 

.96457 

.03674 

2 

59 

.86878 

1.15104 

.89988 

1.11126 

.93197 

1.07299 

.96513 

1.03613 

1 

60 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

.96569 

1.03553 

0 

t 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

f 

49° 

48» 

1           47° 

46° 

TABLE  II.      TANGENTS  AND   COTANGENTS. 


1    / 

4 

£0 

t 

/ 

4 

t4° 

f 

4 

40 

/ 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.96569 

1.03553 

60 

~20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

1 

.96625 

1.03493 

59 

21 

.97756 

.02295 

39 

41 

.98901 

1.01112 

19 

2 

.96681 

1.03433 

58 

22 

.97813 

.02236 

38 

42 

.98958 

1.01053 

18 

3 

.96738 

1.03372 

57 

23 

.97870 

.02176 

37 

43 

.99016 

1.00994 

17 

4 

.96794 

1.03312 

56 

24 

.97927 

.02117 

36 

44 

.99073 

1.00935 

16 

5 

.96850 

1.03252 

55 

25 

.97984 

.02057 

35 

45 

.99131 

1.00876 

15 

6 

.96907 

1.03192 

54 

26 

.98041 

.01998 

34 

46 

.99189 

1.00818 

14 

7 

.96963 

1.03132 

53 

27 

.98098 

.01939 

33 

47 

.99247 

1.00759 

13 

8 

.97020 

1.03072 

52 

28 

.98155 

.01879 

32 

48 

.99304 

1.00701 

12 

9 

.97076 

1.03012 

51 

29 

.98213 

.01820 

31 

49 

.99362 

1.00642 

11 

10 

.97133 

1.02952 

50 

30 

.98270 

.01761 

30 

50 

.99420 

1.00583 

10 

11 

.97189 

1.02892 

-49 

31 

.98327 

1.01702 

29 

51 

.99478 

1.00525 

9 

12 

.97246 

1.02832 

48 

32 

.98384 

1.01642 

28 

52 

.99536 

.00467 

8 

13 

.97302 

1.02772 

47 

33 

.98441 

1.01583 

27 

53 

.99594 

.00408 

7 

14 

.97359 

1.02713 

46 

34 

.98499 

1.01524 

26 

54 

.99652 

.00350 

6 

15 

.97416 

1.02653 

45 

35 

.98556 

1.01465 

25 

55 

.99710 

.00291 

5 

16 

.97472 

1.02593 

44 

36 

.98613 

1.01406 

24 

56 

.99768 

.00233 

4 

1? 

.97529 

1.02533 

43 

37 

.98671 

1.01347 

23 

57 

.99826 

.00175 

3 

18 

.97586 

1.02474 

42 

38 

.98728 

1.01288 

22 

58 

.99884 

1.00116 

2 

19 

.97643 

1.02414 

41 

39 

.98786 

1.01229 

21 

59 

.99942 

1.00058 

1 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

60 

1.00000 

1.00000 

0 

/ 

Cotang 

Tang 

/ 

/ 

Cotang 

Tang 

/ 

Cotang 

Tang 

t 

4 

5° 

4 

5° 

4 

5« 

LENGTHS   OF  CIRCULAR  ARCS. 
Radius  =  1. 


Degrees. 

Minutes. 

Seconds. 

2 
8 

5 

0.017  453  293 
.034  906  585 
.052  359  878 
.069  813  170 
.087  266  463 

0.000  290  888 
.000  581  776 
.000  872  664 
.001  163553 
.001  454  440 

0.000  004  848 
.000  009  695 
.000  014  544 
.000  019  393 
.000  024  241 

6 

7 

10 

.104719  755 
.122  173  048 
.139626340 
.157  079  633 
.174532925 

.001  745  329 
.002036217 
.002  327  106 
.002617994 
.002  908  882 

.000  029  089 
.000  033  937 
.000  038  785 
.000043633 
.000  048  481 

152 


TABLE   III.      MAGNETIC   NEEDLE. 


TABLE  III. 

DAILY  VARIATION  OF  THE  MAGNETIC  NEEDLE  AT 
PHILADELPHIA,    PA. 


fe 

o3 

* 

be 

P- 

"5 

o 

1 

oS 

o> 

o3 

P- 

EC, 

S 

* 

§ 

1-5 

^ 

< 

£ 

o 

* 

Q 

6A.M. 

+0.6 

+1.2 

+1.8 

+2.6 

+3.7 

-f3.9 

+4.2 

+4.7 

+3.5 

+1.3 

+1.2 

+0.7 

7 
8 

+1.2 
+•21 

+1.9 

+2.5 

+2.9 

+3.7 

+3.5 

+4.0 

+4.7 
+4.7 

+5.0 
+5.1 

+5.4 
+5  4 

+5.7 
+5.5 

+4.5 
+4.5 

+  1.7+1.7 
+2.2+1.9 

+1.0 

+1.4 

9 

10 

+2.5 
+1.6 

+25 
+1.5 

+3.4 
+1.8 

+3.4 
+1.5 

-(-3.2 
+0.8 

+3.8 
+1.2 

+4.0 
+1.5 

+3.7 
+0.6 

+2.8 
-0.1 

+1.9+1.5 
+0.8+0  4 

+1.6 
-fl.l 

11 

-o.a 

-0.2 

-0.6 

-1.1 

-1.9 

-1.7 

-1.5 

-2.1J 

-3.2 

—  0  8!  —  1  1 

-0.3 

Noon 

-2.3 

-2  0 

-2.7 

-3.6 

-4.1 

-4.0 

-3.9 

-5.4 

-5.2 

-2.6  -2.3 

-1.9 

1 

-3.4 

-3.0 

-3.9 

-5.1 

-5.1 

-5.0 

-5.3 

-6.3 

-5.5 

-3.2  -2.8 

-3.0 

2 

-3.3  -3.0 

-3.9 

-5.2 

-4.fl 

-4.8 

-5.4 

-5.5 

-4.5 

-3.0 

-2.6 

-3.0 

ft 

-2.5  -2.4 

-3.2 

-4.3 

-3.9 

-3.8 

-4.5 

-3.8 

-3.0 

-2.2  -1.9 

-2.3 

4 

-1.5 

-1.7 

o  g 

-3.0 

-2.5 

-2.6 

-3.3 

-2.0 

-1.7 

—  1  1  —1  2 

-1.3 

5 

-0.9  -1.2 

-1.6 

-1.8 

-1.2 

-1.6 

—  2  C 

-O.S 

-O.fl 

-0.3  -0.6 

-0.6 

6 

-0.6 

-0.8 

-1.0 

-0.9 

-0.4 

-0.9 

-1.2 

-0.5 

-0  3 

+0.4 

-0.1 

-0.1 

The  above  table,  which  is  taken  from  the  U.  S.  Coast  and 
Geodetic  Survey  Report  for  1881,  gives  the  mean  results  of  five 
years'  observations  of  the  daily  variation  of  the  magnetic  needle 
at  Philadelphia.  A  plus  sign  indicates  a  deviation  of  the  north 
end  of  the  needle  to  the  eastward  of  the  magnetic  meridian,  a 
minus  sign  indicates  a  deviation  to  the  westward. 

For  other  places  in  the  United  States  the  daily  variation  may 
be  approximately  ascertained  by  multiplying  the  values  for 
Philadelphia  by  the  numbers  taken  from  the  following  supple- 
mentary table.  For  example,  at  a  place  in  latitude  45  degrees 


Lat. 

Long. 

70°. 

Long. 
80°. 

Long. 
90». 

Long. 
100°. 

Long. 
110°. 

Long. 
120°. 

25° 

0  64 

0  64 

0  63 

0  60 

30 

0  71 

0  70 

0  68 

0.66 

0  65 

35 
40 
45 
50 

0  .93 
1.05 
1.31 

0.86 
1.00 
1.35 

0.80 
0.93 
1.20 
1.50 

0.77 

0.90 
1  05 
1.67 

0.76 
0.82 
0  95 
1.24 

0.74 

0.80 
0.93 
1.14 

and  longitude  95  degrees  the  multiplier  is  1. 13.  In  southern 
latitudes,  moreover,  the  maximum  deviations  occur  about  an 
hour  later  than  in  northern,  and  in  any  particular  case  the 
table  cannot  be  depended  upon  within  one  hour  on  account  of 
minor  irregularities  and  disturbances. 


TABLE   IV.       DEGREES   AND   TIME. 


TO  REDUCE  DEGREES  TO  TIME. 

TO  REDUCE  TIME  TO  DEGREES. 

0 

H.  M. 

o 

H.  M. 

CO 

« 

8 

M. 

9     / 

M. 

0     , 

/ 

M.  S. 

/ 

M.S. 

I 

73   ,2> 
1   § 

| 

1 

S. 

/    // 

S. 

/  // 

// 

S.  T. 

II 

S.  T. 

I 

w  S 

W 

a 

T. 

//  /// 

T. 

//  /// 

1 

0   4 

51 

3  24 

101 

6  44 

1 

15 

1 

0  15 

51 

1245 

2 

0  8 

52 

3  28 

102 

6  48 

14 

224 

2 

0  30 

52 

13  0 

3 

0  12 

53 

3  32 

103 

6  52 

2 

30 

3 

0  45 

53 

13  15 

4 

0  16 

54 

3  36 

104 

6  56 

24 

374 

4 

1  0 

54 

13  30 

5 

0  20 

55 

3  40 

105 

7  0 

3 

45 

5 

1  15 

55 

13  45 

6 

0  24 

56 

3  44 

106 

7  4 

3J 

524 

6 

1  30 

56 

14  0 

7 

0  28 

57 

3  48 

107 

7  8 

4 

60 

7 

1  45 

57 

14  15 

8 

0  32 

58 

3  52 

108 

7  12 

44 

674 

8 

2  0 

58 

14  30 

9 

0  36 

59 

3  56 

109 

7  16 

5 

75 

9 

2  15 

59 

14  45 

10 

0  40 

60 

4  0 

110 

720 

54 

824 

10 

2  30 

60 

15  0 

11 

0  44 

61 

4   4 

115 

7  40 

6 

90 

11 

2  45 

61 

15  15 

12 

0  48 

62 

4  8 

120 

8  0 

64 

974 

12 

3  0 

62 

15  30 

13 

0  52 

63 

4  12 

125 

8  20 

7 

105 

13 

3  15 

63 

15  45 

14 

0  56 

64 

4  16 

130 

8  40 

7* 

112J 

14 

3  30 

64 

16  0 

15 

1   0 

65 

4  20 

135 

9  0 

8 

120 

15 

3  45 

65 

16  15 

16 

4 

66 

4  24 

140 

9  20 

84 

1274 

16 

4  0 

66 

16  30 

17 

8 

67 

4  28 

145 

9  40 

9 

135 

17 

4  15 

67 

16  45 

18 

IS 

68 

4  32 

150 

10  0 

94 

1424 

18 

4  30 

68 

17  0 

19 

16 

69 

4  36 

155 

10  20 

10 

150 

19 

4  45 

69 

17  15 

20 

20 

70 

4  40 

160 

10  40 

104 

1574 

20 

5  0 

70 

17  30 

21 

24 

71 

4  44 

165 

11  0 

11 

105 

21 

5  15 

71 

1745 

22 

28 

72 

4  48 

170 

11  20 

114 

1724 

22 

5  30 

72 

18  0 

23 

32 

73 

4  52 

175 

11  40 

12 

180 

23 

5  45 

73 

18  15 

24 

36 

74 

4  56 

180 

12  0 

124 

1874 

24 

6  0 

74 

18  30 

25 

1  40 

75 

5   0 

185 

12  20 

13 

195 

25 

6  15 

75 

18  45 

26 

1  44 

76 

5   4 

190 

12  40 

134 

202* 

26 

6  30 

76 

19  0 

27 

1  48 

77 

5   8 

195 

13  0 

14 

210 

27 

6  45 

77 

19  15 

28 

1  52 

78 

5  12 

200 

13  20 

144 

2174 

28 

7  0 

78 

19  30 

29 

1  56 

79 

5  16 

205 

1340 

15 

225 

29 

7  15 

79 

19  45 

30 

2   0 

80 

5  20 

210 

14  0 

154 

2324 

30 

7  30 

80 

20  0 

31 

2   4 

81 

5  24 

215 

14  20 

16 

240 

31 

745 

81 

20  15 

32 

2   8 

82 

5  28 

220 

14  40 

164 

2474 

32 

8  0 

82 

20  30 

33 

2  12 

83 

5  32 

225 

15  0 

17 

255 

33 

8  15 

83 

20  45 

34 

2  16 

84 

5  36 

230 

15  20 

174 

2624 

34 

8  30 

84 

21  0 

35 

2  20 

85 

5  40 

235 

15  40 

18 

270 

35 

8  45 

85 

21  15 

36 

2  24 

86 

5  44 

240 

16  0 

184 

277i 

36 

9  0 

86 

21  30 

37 

2  28 

87 

5  48 

245 

16  20 

19 

285 

37 

9  15 

87 

21  45 

38 

2  32 

88 

5  52 

250 

16  40 

194 

2924 

38 

9  30 

88 

22  0 

39 

2  36 

89 

5  56 

255 

17  0 

20 

300 

39 

9  45 

89 

22  15 

40 

2  40 

90 

6   0 

260 

17  20 

204 

3074 

40 

10  0 

90 

2230 

41 

2  44 

91 

6   4 

270 

18  0 

21 

315 

41 

10  15 

91 

22  45 

42 

2  48 

92 

6   8 

280 

18  40 

214 

3224 

42 

10  30 

92 

23  0 

43 

2  52 

93 

6  12 

290 

19  20 

22 

330 

43 

10  45 

93 

23  15 

44 

2  56 

94 

6  16 

300 

20  0 

224 

3374 

44 

11  0 

94 

23  30 

45 

3   0 

95 

6  20 

310 

20  40 

23 

345 

45 

11  15 

95 

23  45 

46 

3   4 

96 

6  24 

320 

•  21  20 

234 

3524 

46 

11  30 

96 

24  0 

47 

3   8 

97 

6  28 

330 

22  0 

24 

360 

47 

11  45 

97 

24  15 

48 

3  12 

98 

6  32 

340 

22  40 

48 

12  0 

98 

24  30 

49 

3  16 

99 

6  36 

350  |  23  20 

49 

12  15 

99 

24  45 

50 

3  20 

100 

6  40 

360 

34  0 

50 

12  30 

100 

25  0 

154 


TABLE  V.      POLARIS. 


TABLE  V. 

LOCAL  TIMES  OF  ELONGATIONS    OF  POLARIS  IN  1915. 
For  40°  North  Latitude  and  90°  West  Longitude. 


Date  in  1915. 

Eastern  Elongation. 

Western  Elongation. 

h.                m. 

h.              m. 

January         1 

12       51.7  P.M. 

12       42.1 

.M 

15 

11       52.5 

.M. 

11       46.8 

.M 

February       1 

10       45.3 

.M. 

10       39.7 

.M 

15 

9       50.1 

.M. 

9       44.4 

.M 

March            1 

8       54.8 

.M. 

8       49.2 

.M 

15 

7       59.6 

.M. 

7       54.0 

.M 

April              1 

6       52.7 

.M. 

6       47.1 

.M 

15 

5       57.7 

.M. 

5       52.0 

.M 

May               1 

4       54.8 

.M. 

4       49.2 

.M 

15 

3       59.9 

.M. 

3       54.2 

.M 

June                1 

2       53.3 

.M. 

2       47.6 

.M 

15 

1       58.5 

.M. 

1       52.8 

.M 

July                1 

12       55.9 

.M. 

12       50.2 

.M 

15 

12       01.1 

.M. 

11       51.5 

.M 

August           1 

10       54.5 

.M. 

10       44.9 

.M 

15 

9       59.8 

.M.    * 

9       50.2 

.M 

September     1 

8       53.2 

.M. 

8       43.6 

.M 

15 

7       58.3 

.M. 

7       48.7 

.M 

October         1 

6       55.5 

.M. 

6       45.9 

.M 

15 

6       00.6 

.M. 

5       51.0 

.M 

November     1 

4       53.7 

.M. 

4       44.1 

.M 

15 

3       58.6 

.M. 

3       49.0 

.M 

December      1 

2       55.6 

.M. 

2       46.0 

.M 

15 

2       00.4 

.M. 

1       50.8 

.M 

For  other  years  than  1915, 'the  following  quantities  should 
be  added  or  subtracted  to  the  above  tabular  values: 

For  1913  subtract  2.9  minutes 

1914  subtract  1.5 

1916,  before  March  1,  add          1.6 
1916,  after     Feb.  29,    subtract  2.3 

1917  subtract  0.7 

1918  add  0.9 

1919  add  2.5 
1920,  before  March  1,  add          4.0 
1920,  after    Feb.  29,  add          0.1 

1921  add  1.6 

1922  add  3.1 

1923  add  4.5 
1924,  before  March  1,  add          5.9 
1924,  after    Feb.  29,    add         2.0 

1925  add  3.3 

1926  add  4.6 

1927  '  add  5.9 


TABLE  Y.      POLARIS.  155 

To  obtain  the  time  of  elongation  for  any  day  not  given  in 
the  table,  add  3.93  minutes  for  every  day  from  it  to  the  day 
of  the  next  following  tabular  value.  For  example,  the 
eastern  elongation  on  Nov.  12,  1915,  occurred  at  4h  10m.4  P.M. 
in  latitude  40°  and  longitude  90°. 

For  any  latitude  other  than  40°,  between  25°  and  50°  north, 
there  should  be  added  to  the  time  of  western  elongation  0.10 
minutes  for  every  degree  south  of  40°  and  0.16  minutes  be 
subtracted  for  every  degree  north  of  40°.  For  eastern  elon- 
gations 0.10  minutes  should  be  subtracted  for  every  degree 
south  of  40°  and  0.16  minutes  be  added  for  every  degree 
north  of  40°.  For  any  longitude  other  than  90°  west  of 
Greenwich,  add  0.16  minutes  for  each  15  degrees  east  of  the 
ninetieth  meridian  and  subtract  0.16  minutes  for  each  15  de- 
grees west  of  that  meridian. 

The  time  in  Table  V  is  local  time,  which  is  the  same  as 
mean  solar  time.  Local  time  can  be  reduced  to  standard 
time  by  adding  or  subtracting  4.0  minutes  for  each  degree  of 
longitude  west  or  east  of  the  meridian  of  the  standard. 

As  an  example  involving  all  these  corrections,  let  it  be 
required  to  find,  for  an  observer  in  north  latitude  42°  06'  and 
west  longitude  78°  45',  the  standard  time  of  the  eastern  elon- 
gation of  Polaris  on  Aug.  28,  1920.  From  the  Table  the  local 
time  8h  35m.2  P.M.  is  found  for  Sept.  1,  1915,  and  to  this  is 
added  the  correction  for  1920,  making  8h  53m.3  P.M.  for  Sept. 
1,  1920.  To  this  15^.7  are  added  for  the  four  days  from 
Aug.  28  to  Sept.  1,  giving  9h  09m.O  P.M.  for  Aug.  24,  1920. 
The  corrections  for  latitude  and  longitude  of  the  given  sta- 
tion are  —  Om.34  and  +0m.12;  hence  the  eastern  elongation 
will  occur  at  that  station  on  Aug.  28,  1920,  at  9h  08m.8  P.M. 
On  a  watch  indicating  eastern  standard  time  the  time  of  the 
eastern  elongation  for  the  given  day  and  station  will  be 
9h  23m.8  P.M.  A  result  deduced  in  this  manner  will  usually 
be  correct  within  about  Om.3.j 

Table  V  has  been  taken  from  "Principal  Facts  of  the 
Earth's  Magnetism,"  issued  in  1914  by  the  U.  S.  Coast  and 
Geodetic  Survey. 


156  TABLE   VI.      POLARIS. 

TABLE  VI. 

AZIMUTHS   OF  POLAKIS  AT  ELONGATION. 


Lat. 

1911 

1912 

1913 

1914 

1915 

1916 

1917 

1918 

25° 

1°  17'  .4 

1°  17'.  0 

1°  16'  .7 

1°  16'  .4 

1°  16'  .0 

lo  15'  .7 

1°  15'  .3 

1°  15'  .0 

26 

18  .0 

17  .7 

17  .3 

17  .0 

16  .6 

16  .3 

16  .0 

15  .6 

27 

18  .7 

18  .4 

18  .0 

17  .7 

17  .3 

17  .0 

16  .6 

16  .3 

28 

19  .4 

19  .1 

18  .7 

18  .4 

18  .0 

17  .7 

17  .3 

17  .0 

29 

20  .2 

19  .8 

19  .5 

19  .1 

18  .8 

18  .4 

18  .1 

17  .7 

30 

21  .0 

20  .6 

20  .3 

19  .9 

19  .6 

19  .2 

18  .8 

18  .5 

31 

21  .8 

21  .5 

21  .1 

20  .7 

30  .4 

20  .0 

19  .? 

19  .3 

32 

2-^  .7 

22  .3 

22  .0 

21  .6 

21  .2 

20  .9 

20  .5 

20  .1 

33 

23  .6 

23  .3 

22  .9 

22  .5 

22  .1 

21  .8 

21  .4 

21  .0 

34 

24  .6 

24  .2 

23  .8 

23  .5 

23  .1 

22  .7 

22  .4 

22  .0 

35 

25  .G 

25  .2 

24  .9 

24  .5 

24  .1 

23  .7 

23  .3 

23  .0 

36 

26  .7 

26  .3 

26  .9 

25  .5 

25  .2 

24  .8 

24  .4 

24  .0 

37 

27  .8 

27  .4 

27  .0 

26  .7 

26  .3 

25  .9 

25  .5 

25  .1 

38 

29  .0 

28  .6 

28  .2 

27  .8 

27  .4 

27  .0 

26  .6 

26  .2 

39 

30  .2 

29  .8 

29  .4 

29  .0 

28  .6 

28  .2 

27  .8 

27  .5 

40 

31  .6 

31  .1 

30  .7 

30  .3 

29  .9 

29  .5 

29  .1 

28  .7 

41 

32  .9 

32  .5 

32  .1 

31  .7 

31  .3 

30  .9 

30  .4 

30  ,0 

4:2 

34  .4 

34  .0 

33  .5 

33  .1 

32  .7 

32  .3 

31  .9 

31  .5 

43 

35  .9 

35  .5 

35  .0 

34  .6 

34  .2 

33  .8 

33  .4 

32  .9 

44 

37  .5 

37  .1 

36  .6 

36  .2 

35  .8 

35  .3 

84  .9 

34  .5 

45 

39  .2 

38  .7 

38  .3 

37  .8 

37  .4 

37  .0 

36  .6 

36  .1 

46 

41  .0 

40  .5 

40  .1 

39  .0 

39  .2 

38  .7 

38  .3 

37  .8 

47 

42  .8 

42  .4 

41  .9 

41  .5 

41  .0 

40  .6 

40  .1 

39  .7 

48 

44  .8 

44  .4 

43  .9 

43  .4 

43  .0 

42  .5 

42  .0 

41  .6 

49 

46  .9 

46  .4 

46  .0 

45  .5 

45  .0 

44  .5 

44  .1 

43  .6 

50 

49  .1 

1  48  .6 

1  48  .2 

1  47  .7 

1  47  .2 

1  46  .7 

1  46  .2 

1  45  .7 

The  azimuths  in  Table  VI  are  reckoned  from  the  true  north 
toward  the  east  for  eastern  elongation  and  from  the  true  north 
toward  the  west  for  western  elongation.  For  intermediate 
latitudes  values  may  be  obtained  by  interpolation;  for 
example,  in  latitude  41°  30'  the  mean  azimuth  during  1913  is 
1°  32'.8,  and  for  July  1,  1913,  the  azimuth  is  1°  33'.2.  An 
azimuth  deduced  in  this  manner  will  in  general  be  correct 
within  0'.3 

This  table  has  been  taken  from  "  Principal  Facts  of  the 
Earth's  Magnetism,"  issued  in  1914  by  the  U.  S.  Coast  and 
Geodetic  Survey. 


TABLE  VI.     POLARIS. 
AZIMUTHS  OF  POLARIS  AT  ELONGATION. 


157 


Lat. 

1919 

1920 

1921 

1922 

1923 

1924 

1925 

1926 

~25° 

1°  14'.  7 

1°  14'.  7 

1°  14'.  0 

1°  13'.  6 

1°  13'.  3 

1°  13'.  0 

1°  12'.  6 

1°  12'.  3 

26 

15.3 

14  .9 

14.7 

14.2 

13.9 

13  .6 

13.2 

12.9 

27 

15.9 

15.6 

15.2 

14.9 

14.6 

14.2 

13.9 

13.5 

28 

16.6 

16.3 

15.9 

15.6 

15.2 

14  .9 

14.6 

14.2 

29 

17.4 

17.0 

16.6 

16.3 

16.0 

15.6 

15.2 

14.9 

30 

19.1 

18.8 

17.4 

17.0 

16.7 

16.4 

16.0 

15.6 

31 

19  .9 

18.6 

18.2 

17.9 

17.5 

17  .2 

16  .8 

16.4 

32 

19.8 

18.4 

19.1 

18.7 

18.3 

18.0 

17.6 

17  .2 

33 

20.7 

20.3 

19.9 

19.6 

19.2 

18.8 

18.5 

18.1 

34 

21.  6 

21.2 

20.9 

20.5 

20.1 

19.8 

19.4 

19.0 

35 

22.6 

22.  2 

21.  8 

21.5 

21.1 

20.7 

20.4 

20.0 

36 

23  .6 

23  .3 

22.  9 

22.5 

22.1 

21.7 

21.4 

21.0 

37 

24.7 

24.3 

24.0 

23.6 

23.2 

22  .8 

22.4 

22.0 

38 

25.9 

25  .5 

25.1 

24  .7 

24  .3 

23.9 

23.5 

23  .2 

39 

27.1 

26.7 

26.3 

25.8 

25.5 

25.1 

24.7 

24.3 

40 

28.3 

27.9 

27.5 

27.1 

26.7 

26.3 

25.9 

25.5 

41 

29.6 

29.1 

28.8 

28.4 

28  .0 

27.6 

27.2 

26.8 

42 

31.0 

30.6 

30.2 

29.8 

29.4 

29.0 

28  .6 

28.2 

43 

32.  5 

32.1 

31  .8 

31.2 

30  .8 

30.4 

30.0 

29.6 

44 

34.1 

33.  6 

33.2 

32.8 

32.4 

31.9 

31.5 

31.1 

45 

35.7 

35.3 

34.8 

34.4 

34.0 

33.5 

33.1 

32.6 

46 

37.4 

37.0 

36.5 

36.1 

35.6 

35  .2 

34  .8 

34.3 

47 

39.2 

38  .8 

38  .3 

37  .9 

37  .4 

36.5 

36.5 

36.1 

48 

41.1 

40.7 

40.2 

39.8 

39.3 

'   38.8 

38  .4 

37.9 

49 

43.1 

42  .7 

42  .2 

41.7 

41  .3 

40.8 

40.3 

39.9 

50° 

1°  45'.3 

1°  44'.  8 

1°  44'.  3 

1°  43'.  8 

1°  43'.  4 

1°  42'.  9 

1°  42'.  4 

1°  41'.9 

When  an  azimuth  is  required  with  a  precision  less  than  one 
minute,  a  correction  taken  from  the  following  supplementary 
table  should  be  applied.  For  example,  the  azimuth  as  seen  in 
latitude  42°  on  Dec.  1,  1920,  is  1  °  29'.9.  An  azimuth  deduced 
in  this  manner  will  generally  be  correct  within  0'.3. 


For  middle  of 

Correction. 

For  middle  of 

Correction. 

January.  .  .  . 

—  0  .5 

July 

+0  .2 

February  

-0  .4 

August  ,  .  .  .  . 

+  0  .1 

March  

—  0  .3 

September 

-0  .1 

April. 

0  0 

October 

—  0  4 

May  

+0  .1 

November      

—  0  .6 

a  June  

-|-0  2 

December 

—  0  8 

158 


TABLE   VII.      LINEAR  MEASURES. 


CONVERSION  OF  ENGLISH  INCHES  INTO  CENTIMETRES. 

Ins. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

0 

0.000 

2.5401     5.080 

7.620 

10.16 

12.70 

15.24 

17.78 

20.32 

22.86 

10 

25.40 

27.94 

30.48 

33.02 

35.56 

38.10 

40.64 

43.18 

45.72 

48.26 

20 

50.80 

53.34 

55.88 

58.42 

60.96 

63.50 

66.04 

68.58 

71.12 

73.66 

30 

76.20 

78.74 

81.28 

83.82 

86.36 

88.90 

91.44 

93.98 

96.52 

99.06 

40 

101.60 

104.14 

106.68 

109.22 

111.76 

114.30 

116.84 

119.38 

121.92 

124.46 

50 

127.00 

129.54 

132.08 

134.62 

137.16 

139.70 

142.24 

144.78 

147.32 

149.86 

60 

152.40 

154.94 

157.48 

160.02 

162.56 

165.10 

167.64 

170.18 

172.72 

175  26 

70 

177.80 

180.34 

182.88 

185.42 

187.96 

190.50 

193.04 

195.58 

198.12 

200.96 

80 

203.20 

205.74 

208.28 

210.82 

213.36 

215.90 

218.44 

220.98 

223.52 

226.06 

90 

228.60 

231.14 

233.68 

236.22 

238.76 

241.30 

243.84 

246.38 

248.92 

251.46 

100 

254.00 

256.54 

259.08 

261.62 

264.16 

266.70 

269.24 

271.78274.32 

276.86 

CONVERSION  OF  CENTIMETRES  INTO  ENGLISH  INCHES. 

Cm. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Ins. 

Ins. 

Ins.       Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

0 

0.000 

0.394 

0.787     1.181 

1.575 

1.969 

2.362 

2.75.6 

3.150 

3.543 

10 

3.937 

4.331 

4.742     5.118 

5.512 

5.906 

6.299 

6.693 

7.087 

7.480 

20 

7.874 

8.268 

8.662     9.055 

9.449 

9.843 

10.236  10.630 

11.024 

11.418 

30 

11.811 

12.205 

12.599    12.992 

13.386 

13.780 

14.173,14.567 

14.961 

15.355 

40 

15.748 

16.142 

16.536    16.929 

17.323 

17.717 

18.111  18.504 

18.898 

19.292 

50 

19.685 

20.079 

20.473    20.867 

21.260 

21.654 

22.04822.441 

22.835 

23.229 

60 

23.622 

24.016 

24.410    24.804 

25.197 

25.591 

25.98526.378 

26.772 

27.166 

70 

27.560 

27.953 

28.347    28.741 

29.134 

29.528 

29.92230.316 

30.709 

31.103 

80 

31.497 

31.890 

32.284    32.678 

33.071 

33.465 

33.85934.253 

34.646 

35.040 

90 

35.434 

35.827 

36.221    36.615 

37.009 

37.402 

37.79638.190 

38.58338.977 

100 

39.370)  39.764 

40.158    40.552 

40.945 

41.339 

41.73342.126 

42.52042.914 

CONVERSION  OF  ENGLISH  FEET  INTO  METRES. 

Feet. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Met. 

Met. 

Met. 

Met. 

Met. 

Met." 

Met. 

Met. 

Met. 

Met. 

0 

0.000 

0.3048 

0.6096 

0.9144 

1.2192 

1.5239 

1.8287 

2.1335 

2.4383 

2.7431 

10 

3.0479 

3.3527 

3.6575 

3.9623 

4.2671 

4.  571  9  14.8767  5.  1815 

5.4863 

5.7911 

20 

6.0959 

6.4006 

6.7055 

7.0102 

7.3150 

7.6198 

7.92468.2294 

8.5342 

8.8390 

30 

9.1438 

9.4486 

9.7534 

10.058 

10.363 

10.668 

10.972111.277 

11.582 

11.887 

40 

12.192 

12.496 

12.801 

13.106 

13.  4M 

13.716 

14.02014.325 

14.630 

14.935 

50 

15.239 

15.544 

15.849 

16.154 

16.459 

16.763 

17.068  17.373 

17.678 

17.983 

60 

18.287 

18.592 

18.897 

19.202 

19.507 

19.811 

20.11620.421 

20.726 

21.031 

70 

21.335 

21.640 

21.945 

22.250 

22.555 

22.859 

23.16423.469 

23.774 

24.079 

80 

24.383 

24.688 

24.993 

25.298 

25.602 

25.907 

26.21226.517 

26.822 

27.126 

90 

27.431 

27.736 

28.041 

28.346 

28.651 

28.955 

29.  260  ;  29.  565 

29.870 

30.174 

100 

30.479 

30.784 

31.089 

31.394 

31.698 

32.003 

32.30832.613 

32.918 

33.222 

CONVERSION  OF  METRES  INTO  ENGLISH  FEET. 

Met. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

0 

0.000 

3.2809 

6.5618    9.8427 

13.123 

16.404 

19.685 

22.966 

26.247 

29.528 

10 

32.809 

36.090 

39.37l|  42.651 

45.932 

49.213 

52.494 

55.775 

59.056 

62.337 

20 

65.618 

68.899 

72.179    75.461 

78.741 

82.02-2 

85.303 

88.584 

91.865 

95.146 

30 

98.427 

101.71 

104.99!  108.27 

111.55 

114  83 

118.11 

121.39 

124.67 

127.96 

40 

131.24 

134.52 

137.80    141.08 

144.36 

147.64 

150.92 

154.20 

157.48 

160.76 

50 

164.04 

167.33 

170  61  j  173.89 

177.17 

180.  45(183.  73 

187.01 

190.29 

193.57 

60 

196.85 

200.13 

203.42    206.70 

209.98 

213.26216.54 

219.82 

223.10 

226.38 

70 

229.66 

232.94 

236.22    239.51 

242.79 

246.07249.35 

252.63 

255.91 

259.19 

80 

262.47 

265.75 

269.03    272.31 

275.60 

278.88282.16 

285.44 

288.72 

292.00 

90 

295.28 

298.56 

301.84!  305.12 

308.40 

311.69:314.  97 

318.25 

321.53 

324.81 

100 

328.09 

331.37 

334.65   337.93 

341.21 

344.49347.78 

351.06 

354.34 

357.  62 

TABLE   VII.      LINEAR  MEASURES. 


159 


CONVERSION  OF 

ENGLISH  STATUTE-MILES  INTO  KILOMETRES. 

Miles. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

0 

0.0000 

1.6093 

3.2186 

4.8279 

6.  4372  i  8.0465 

9.6558 

11.2652 

12.8745 

14.4848 

10 

16.093 

17.70219.312 

20.921 

22.530   24.139 

25.749   27.358 

28.967 

30.577 

20 

32.186 

33.795 

35.405 

37.014 

38.623    40.232 

41.842 

43.451 

45.060 

46.670 

30 

48.279 

49.88851.498 

53.107 

54.7K 

)    56.325 

57.935 

59.544 

61.153 

62.763 

40 
50 

64.  37265.981;  67.  591 
80.  465  82.  074!  83.  684 

69.200 
85.293 

70.80< 
86.905 

)    72.418 
2    88.511 

74.028 
90.121 

75.637 
91  730 

77.246 
93.339 

78.856 
94.949 

60 

96.55898.167 

99  777 

101.39 

102.9 

)    104.60 

106  21 

107.82 

109.43 

111.04 

70 

112.65 

114.26 

115.87 

117.48 

119.01 

*    120.69 

122.30 

123.91 

125.52 

127'.  13 

80 

128.74 

130.35 

131.96 

133.57 

135.1 

r    136.78 

138.39 

140.00 

141.61 

143.22 

90 

144.85 

146.44 

148.05 

149.66 

151.2 

3    152.87 

154.48 

156.09 

157.70 

159.31 

100 

160.93162.53 

164  14 

165  75 

167.3 

3    168.96 

170.57 

172.18 

173.79 

175.40 

CONVERSION  OF 

KILOMETRES  INTO  ENGLISH   STATUTE-MILES. 

Kilom. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Miles. 

Miles. 

Miles. 

Miles. 

Miles 

.  Miles. 

Miles. 

Miles. 

Miles? 

Miles. 

0 

0.0000 

0.6214 

1.2427 

1.8641 

2.485 

5    3.1069 

3.7282 

4.3497 

4.9711 

5.5924 

10 

6.2138 

6.8352 

7.4565 

8.0780 

8.699 

1    9.3208 

9.9421 

10.562 

11.185 

11.805 

20 

12.427 

13  049 

13  670 

14.292 

14.91 

3    15.534 

16.156 

16.776 

17.399 

18.019 

30 

18.641 

19.263 

19.884 

20.506 

21.12 

7    21.748 

23  370 

22.990 

23.613 

24.233 

40 

24.855 

25.477 

26.098 

26.720 

27.34 

1    27.962 

28.584 

29.204 

29.827 

30.447 

50 

31.069 

31.690 

32.311 

32.933 

33.55 

\    34.175 

34.797 

35.417 

36.040 

36.660 

60 

37.282 

37.904 

38.525 

39.147 

39.76 

3    40.389 

41.011 

41.631 

42.254 

42.874 

70 

43.497 

44.118 

44.739 

45.361 

45.98 

2    46.603 

47.225 

47.845 

48.468 

49.088 

80 

49.711 

50.332 

50.953 

51.575 

52.19 

5    52.817 

53.439 

54.059 

54.682 

55.302 

90 

55.924 

56.545 

57.166 

57.788 

58.40 

9    59.030 

59.652 

60.272 

60.895 

61.515 

100 

62.138 

62.759 

63.380 

64.002 

64.62 

3    65.244 

65.866 

66.486 

67.109 

67.729 

TABLE  VIII. 

LENGTH  IN  FEET  OF  1'  ARCS  OF  LATITUDE  AND  LONGITUDE. 

Lat. 

V  Lat. 

I'  Long. 

Lat. 

1'  Lat. 

7  '  Long. 

1° 

6045 

6085 

31fc 

6061 

5226 

2° 

6045 

6083 

32° 

606-> 

5166 

3° 

6045 

6078 

33° 

6063 

5109 

4» 

6045 

6071 

34° 

6064 

5051 

6* 

6045 

6063 

35« 

60G5 

4991 

6° 

6045 

6053 

36» 

6066 

4930 

7° 

6046 

6041 

37° 

6067 

4867 

8° 

6046 

6027 

38° 

6068 

4802 

9° 

6046 

6012 

39° 

6070 

4736 

10° 

6047 

5994 

40° 

6071 

4669 

11° 

6047 

5975 

41* 

6072 

4600 

12* 

6048 

5954 

42° 

6073 

4530 

13° 

6048 

5931 

43« 

6074 

4458 

14° 

6049 

5907 

44° 

6075 

4385 

15° 

6049 

5880 

45° 

6076 

4311 

16° 

6050 

5852 

46° 

6077 

4235 

17° 

6050 

5822 

47° 

6078 

4158 

18° 

6051 

5790 

48° 

6079 

4080 

19° 

6052 

5757 

49° 

6080 

4001 

20° 

6052 

5721 

50° 

6081 

3920 

21° 

6053 

5684 

51° 

6082 

3838 

22° 

6C54 

5646 

52° 

6084 

3755 

23* 

6054 

5605 

53° 

6085 

3671 

24° 

6055 

5563 

54° 

6086 

3586 

25° 

6056 

5519 

55° 

6087 

3499 

26° 

6057 

5474 

56° 

6088 

3413 

27° 

6058 

5427 

57° 

6089 

3323 

28° 

6059 

5378 

58° 

6090 

3233 

29° 

6060 

5327 

59° 

6091 

3142 

30° 

6061 

5275 

•    60° 

6092 

3051 

160 


TABLE   IX.      INCLINED   DISTANCES. 


TABLE  IX. 

REDUCTION  OF  INCLINED  DISTANCES  TO  THE  HORIZONTAL. 
Inclined  Distance  =  100  feet. 


Slope. 

Correction. 

Horizontal 
Distance. 

Slope. 

Correction. 

Horizontal 
Distance. 

0°    00' 

100.000 

8°  00' 

0.973 

99.027 

30 

'"o'.m" 

99.996 

30 

1.098 

98.902 

1      00 

0.015 

99.985 

9    00 

1.231 

98.769 

30 

0  034 

99.966 

30 

1.371 

98.629 

2      00 

0.061 

99.939 

10    00 

1.519 

98.481 

30 

0.095 

99.905 

30 

1.675 

98.325 

3      00 

0.137 

99.863 

11    00 

1.837 

98.163 

30 

0.187 

99.813 

30 

2.008 

97.992 

4      00 

0.244 

99.756 

12    00 

2.185 

97.814 

30 

0.308 

99.692 

30 

2.37'0 

97.630 

5      00 

0.381 

99.619 

13    00 

2.563 

977437 

30 

0.460 

99.540 

30 

2.763 

97.237 

6      00 

0.548 

99.452 

14    00 

2.970 

97.030 

30 

0.643 

99.357 

30 

3.185 

96.815 

7      00 

0.745 

99.255 

15    00 

3  407 

96.593 

30 

0.856 

99.144 

30 

3.637 

96.363 

ANSWERS  TO  PROBLEMS. 

Prob.  1:  A  =  24°  39',  B  —  17°  56'.     Prob.  2  :  azimuth  of  DE 

—  106°  45'.      Prob.    3:   latitude  =  +  2458.2   feet,    longitude 

—  -f-  5379.4  feet.     Prob.  4  :  area  ==  5  acres,  104  rods,  84  square 
feet.     Prob.  5  :  for  BG,  +  382. 1  feet,  and  +  823.3  feet.     Prob. 
6  :  Area  —  11  acres,  116  rods,  126  square  feet.     Prob.  8:   dis- 
tance —  10340  feet.         Prob.  9 :  3/is  226.6  feet  above  N.   Prob. 
10  :  ADD  =  117°  52£',  COD  =  22°  01^.     Prob.  11  :  true  area 
=  7  acres,  146  rods,  222  square  feet.      Prob.  13 :  maximum 
declination  8°  03'  in  January,  1916.     Prob.  14 :  area  =  3  acres,  0 
roods,  4.7  square  rods.     Prob.  18  :  N  78°  06'  W,  26  links,  for  A  ; 
S  74°  35'  W,  56  links  for  G.     Prob.  20  :  476.954  and  477.715 
chains.     Prob.  23  :  error  =  0.025  feet.     Prob.  28  :  pull  =  17.1 
pounds.      Prob.    30 :    latitude  =  2000.000    feet,    longitude  = 
4000.000  feet.     Prob.   31:    83*  feet,  398.6  acres.     Prob.  34: 
902.6  and  417.1  for  the  first  point. 


TABLE  X. 


KEDUCTION  OF  STADIA  HEADINGS 

TO 
HORIZONTAL  DISTANCES 

AND  TO 

DIFFERENCES  OF  ELEVATION. 


This  table  was  computed  by  Professor  Arthur  Winslow, 
State  Geologist  of  Missouri. 


162 


TABLE  X.      STADIA   REDUCTIONS. 


TABLE  X. 

STADIA  REDUCTIONS  FOR  READING  100. 


Minutes. 

0° 

1° 

2° 

3° 

Hor.      Diff. 
Dist.     Elev. 

Hor.       Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

0' 

100.00      .00 

99.97      1.74 

99.88      3.49 

99.73      5.23 

2 

.06 

.80 

99.87      3.55 

99.  73      5.28 

4 

"         .12 

.86 

3.60 

99.71      5.34 

6 

*'         .17 

99.96        .92 

3.66 

"         5.40 

8 

.23 

.98 

99.86      3.72 

99.70      5.46 

10 

.29 

44         2.04 

3.78 

99.69      5.52 

12 

.35 

44         2.09 

99.85      3.84 

44         5.57 

14 

.41 

99.95      2.15 

3.90 

99.68      5.63 

16 

«'         ,47 

44         2.21 

99.84      3.95 

5.69 

18 

.52 

2.27 

4.01 

99.67      5.75 

20 

"         .58 

2.33 

99.83      4.07 

99.66      5.80 

22 

.64 

99.94      2.38 

4.13 

44         5.86 

24 

"         .70 

44         2.44 

99.82      4.18 

99.65      5.92 

26 

99.99      .76 

2.50 

4.24 

99.64      5.98 

28 

.81 

99.93      2.56 

99.81      4.30 

99.63      6.04 

30 

.87 

2.62 

4.36 

44         6.09 

32 

.93 

44         2.67 

99.80      4.42 

99.62      6.15 

34 

.99 

2.73 

4.48 

6.21 

36 

44         .05 

99.92      2.79 

99.79      4.53 

99.61      6.27 

38 

44         .11 

2.85 

41         4.59 

99.60      6.33 

40 

.16 

2.91 

99.78      4.65 

99.59      6.38 

42 

.22 

99.91      2.97 

44         4.71 

6.44 

44 

99.98      .28 

3.02 

99.77      4.76 

99.58      6.50 

46 

.34 

99.90      3.08 

4.82 

99.57      6.56 

48 

44         .40 

3.14 

99.76      4.88 

99.56      6.61 

50 

44          .45 

3.20 

4.94 

6.67 

52 

.51 

99.89      3.26 

99.75      4.99 

99.55      6.73 

54 

.57 

3.31 

99.74      5.05 

99.54      6.78 

56 

99.97      .63 

«*         3.37 

5.11 

99.53      6.84 

58 

.69 

99.88      3.43 

99.73      5.17 

99.52      6.90 

60 

"         .74 

3.49 

5.23 

9951      6.96 

c+/=    .75 
c+f=  1.00 

.75      .01 
1.00    -.01 

.75        .02 
1.00        .03 

.75        .03 
1.00        .04 

.75        .05 
1.00        .06 

C+/=  1.25 

1.25       .02 

1.25         -03 

1.25        .05 

1.25        .08 

TABLE   X.      STADIA    REDUCTIONS. 


163 


TABLE  X. 

STADIA    REDUCTIONS  FOR  READING  100. 


Minutes. 

4° 

5° 

6° 

7° 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

Hor.       Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

0' 
2 
4 
6 
8 
10 

99.51      6.96 
7.02 
99.50      7.07 
99.49      7.13 
99.48      7.19 
99.47      7.25 

99.24      8.68 
99.23      8.74 
99.22      8.80 
99.21      8.85 
99.20      8.91 
99.19      8.97 

98.91    10.40 
98.90    10.45 
98.88    10.51 
98.87    10.57 
98.86    10  62 
98.85    10.68 

98.51    12.10 
98.50    12.15 
98.48    12.21 
88.47    12.26 
98.46    12.32 
98.44    12.38 

12 
14 
16 
18 
20 

99.46      7.30 
7.36 
99.45      7.42 
99.44      7.48 
99.43      7.53 

99.18      9.03 
99.17      9.08 
99.16      9.14 
99.15      9.20 
99.14      9.25 

98.83    10.74 
98.82    10.79 
98.81     10.85 
98.80    10.91 
98.78    10.96 

98.43    12.43 
98.41     12.49 
98.40    12.55 
98.39    12.60 
98.37    12.66 

22 
24 
26 
28 
30 

99.42      7.59 
99.41      7.65 
99.40      7.71 
99.39      7.76 
99.38      7.82 

99.13      9.31 
99.11      9.37 
99.10      9.43 
99.09      9.48 
99.08      9.54 

98.77    11.02 
98.76    11.08 
98.74    11.13 
98.73    11.19 
98.72    11.25 

98.36    12.72 
98.34    12.77 
98.33    12.83 
98.31     12.88 
98.29    12.94 

32 
34 
36 
38 
40 

99.38      7.88 
99.37      7.94 
99.36      7.99 
99.35      8.05 
99.34      8.11 

99.07      9.60 
99.06      9.65 
99.05      9.71 
99.04      9.77 
99.03      9.83 

98.71    11.30 
98  69    11.36 
98.68    11.42 
98.  G7    11.47 
98.65    11.53 

98.28    13.00 
98.27    13.05 
98.25    13.11 
98.24    13.17 
98.22    ,13.22 

42 
44 
46 
48 
50 

99.33      8.17 
99.32      8.22 
99.31      8.28 
99.30      8.34 
99.29      8.40 

99.01      9.88 
99.00      9.94 
98.99    10.00 
98.98    10.05 
98.97    10.11 

98.64    11.59 
98.63    11.64 
98.61     11.70 
98.60    11.76 
98.58    11.81 

98.20    33.28 
98.19    13.33 
98.17    13  39 
98.16    13.45 
98.14    13.50 

52 
54 
56 
58 
60 

99.28      8.45 
99.27      8.51 
99.26      8.57 
99.25      8.63 
99.24      8.68 

98.96    10.17 
98.94    10.22 
98.93    10.28 
98.92    10.34 
98.91    10.40 

98.57    11.87 
9856    11.93 
98.54    11.98 
98.53    12.04 
98.51    12.10 

98.13    13.56 
98.11    13.61 
98.10    13.67 
98.08    13.73 
98.06    13.78 

c+/=    .75 
c+/=  1.00 
c+/  =  1.25 

.75        .06 
1.00        .08 
1.25        .10 

.75        .07 
.99        .09 
1.24        .11 

.75        .08 
.99        .11 
1.24        .14 

.74        .10 
.99        .13 
1.24        .16 

164 


TABLE  X.      STADIA  REDUCTIONS. 


I 
TABLE  X. 

STADIA    REDUCTIONS  FOR  READING  100. 


Minutes. 

8° 

9° 

10° 

11° 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.      Elev. 

Hor.       Diff. 
Dist.     Elev. 

0' 
2 
4 
6 

8 
10 

98.06    13.78 
98.05    13.84 
98.03    13.89 
98.01     13.95 
98.00    14.01 
97.98    14.06 

97.55    15.45 
97.53    15.51 
97.52    15.56 
97.50    15.62 
97.48    15.67 
97.46    15.73 

96.98    17.10 
96.96    17.16 
96.94    17.21 
96.92    17.26 
96.90    17.32 
96.88    17.37 

96.36    18.73 
96  34    18.78 
96.32    18.84 
96.29    18.89 
96.27    18.95 
96.25    19.00 

12 
14 
16 
18 
20 

97.97    14.12 
97.95    14.17 
97.93    14.23 
97.92    14.28 
97.90    14.34 

97.44    15  78 
97.43    15.84 
97.41     15.89 
97.39    15.95 
97.37    16.00 

96.86    17.43 
96.84    17.48 
96.82    17.54 
96.80    17.59 
96.78    17.65 

96.23    19.05 
96.21     19.11 
96.18    19.16 
96.16    19.21 
96.14    19.27 

22 
24 
26 

28 
30 

97.88    14.40 
97.87    14.45 
97.85    14.51 
97.83    14.56 
97.82    14.62 

97.35    16.06 
97.33    16.11 
97.31     16.17 
97.29    16.22 
97.28    16.28 

96.76    17.70 
96.74    17.76 
96.72    17.81 
96.70    17.86 
96.68    17.92 

96.12    19.32 
96.09    19.38 
96.07    19.43 
96.05    19.48 
96.03    19.54 

32 
34 
36 
38 
40 

97.80    14.67 
97.78    14.^3 
97.76    14.79 
97.75    14.84 
97.73    14.90 

97.26    16.33 
97.24    16.39 
97.22    16.44 
97.20    16.50 
97.18    16.55 

96.66    17.97 
96.64     18.03 
96.62    18.08 
96.60    18.14 
96.57    18.19 

96.00    19.59 
95  1)8    19.64 
95.96    19.70 
95.93    19.75 
95.91     19.80 

42 
44 
46 

48 
50 

97.71    1495 
97.69    15.01 
97.68    15.06 
97.66    15.12 
97.64    15.17 

97.16    16.61 
97.14    16.66 
97.12    16.72 
97.10    16.77 
97.08    16.83 

96.55    18.24 
96.53    18.30 
96.51     18.35 
96.49    18.41 
96.47    18.46 

95.89    19.86 
95.86    19.91 
95.84    19.96 
95.82    20.02 
95.79    20.07 

52 
54 
56 
58 
60 

97.62    15.23 
97.61     15.28 
97.59    15.34 
97.57    15.40 
97.55    15.45 

97.06    16.88 
97.04     16.94 
97.02    16.99 
97.00    17.05 
96.98    17.10 

96.45    18.51 
96.42    18.57 
96.40    18.62 
96.38    18.68 
96.36    18.73 

95.77    20.12 
95.75    20.18 
95.72    20.23 
95.70    20.28 
95.68    20.34 

.73        .15 
.98        .20 
1.22        .25 

c4-/  =    .75 
c4-/=1.00 
c+/=1.25 

.74        .11 
.99        .15 
1.23        .18 

.74        .12 
.99        .16 
1.23        .21 

.74        .14 
.98        .18 
1.23        .23 

TABLE  X.      STADIA  REDUCTIONS. 


165 


TABLE  X. 

STADIA  KEDUCTIONS  FOR  READING  100. 


Minutes. 

12° 

13° 

14° 

15° 

Hor.      Diff. 
Dist.     Elev. 

Hor.       Diff. 
Dist.     Elev. 

Hor.       Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist,     Elev. 

0' 
2 
4 
6 

8 
10 

95.68    20.34 
95.65    20.39 
95.63    20.44 
95.61    20.50 
95-58    20.55 
95.56    20.60 

94.94    21.92 
94.91    21.97 
94.89    22.02 
94.86    22.08 
94.84    22.13 
94.81    22.18 

94.15    23.47 
94.12    23.52 
94.09    23.58 
94  07    23.63 
94  04    23.68 
94.01    23.73 

93.30    25.00 
93.27    25.05 
93.24    25.10 
93.21    25.15 
93.18    25.20 
93.16    25.25 

12 
M^ 

16 
18 
20 

95.53    20.66 
C5.51    20.71 
95.49    20.76 
95.46    20.81 
95.44    20.87 

94.79    22.23 
94.76    22.28 
94.73    22.34 
94.71    22.39 
94.68    22.44 

93  98    23.78 
93.95    23.83 
93.93    23.88 
93.90    23.93 
93.87    23.99 

93.13    25.30 
93.10    25.35 
93.07    25.40 
93.04    25.45 
93.01    25.50 

22 
24 
26 
28 
30 

95.41     20.92 
95.39    20.97 
95.36    21.03 
95.34    21.08 
95.32    21.13 

94.66    22.49 
94.63    22.54 
94.60    22.60 
94.58    22.65 
94.55    22.70 

93.84    24.04 
93.81    24.09 
93.79    24.14 
93.76    24.19 
93.73    24.24 

92.98    25.55 
92.95    25  60 
92  92    25.65 
92.89    25.70 
92.86    25.75 

3-2 
34 
36 
38 
40 

95.29    21.18 
95.27    21.24 
95.24    21.29 
95.22    21  34 
95.19    21.39 

94.52    22.75 
94.50    22.80 
1)4.47    2-3.85 
94.44    22.91 
94.42    22.96 

93.70    24.29 
93.67    24.34 
93.65    24.39 
93.62    24.44 
93.59    24.49 

9283    25.80 
92.80    25  85 
92.77    25.90 
92.74    25.95 
92.71    26.00 

42 
44 

46 

48 
50 

95.17    21.45 
95.14    21.50 
95.12    21.55 
95.09    21.60 
95.07    21.66 

94.39    23  01 
94.36    23.06 
94  34    23.11 
94.31    23.16 
94.28    23.22 

93.56    24.55 
93.53    24.60 
93.50    24.65 
93.47    24.70 
93.45    24.75 

92  68    26.05 
9-3.65    26.10 
92.62    26.15 
9-3.59    26.20 
92.56    26.25 

52 
54 
56 

58 
60 

95.04    21.71 
95.02    21.76 
94.99    21.81 
94.97    21.87 
94.94    21.92 

94.26    23.27 
94.23    23.32 
94.20    23  37 
94.17    23.42 
94.15    23.47 

.73        .17 
.97        .23 
1.21        .29 

93.42    24.80 
93.39    24.85 
93.36    24.90 
93.33    24.95 
93.30    25.00 

.73        .19 
.97        .25 
1.21        .31 

92.53    26.30 
92.49    26.35 
92.46    26.40 
92.43    26.4o 
92.40    26.50 

c-f/=    .75 
cf  /=  1.00 
C+/=1.25 

.73        .16 
.98        .22 
1.22        .27 

.72        .20 
.96        .27 
1.20        .84 

166 


TABLE  X.      STADIA  REDuuiiOKS. 


TABLE  X. 

STADIA  REDUCTIONS  FOR  READING  100. 


Minutes. 

16° 

17° 

18° 

19° 

Hor.       Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

H-ir.      Diff. 
Dist.     Elev. 

0' 
2 
4 
6 
8 
10 

92.40    26.50 
92.37    26.55 
92.34    26.59 
92.31     26.64 
92.28    26.69 
92.25    26.74 

91.45    27.96 
91.42    28.01 
91.39    28.06 
91.35    28.10 
91.32    28.15 
91.29    28.20 

90.45    29.39 
90.42    29.44 
90.38    29.48 
90.35    29.53 
90.31    29.58 
90.28    29.62 

89.40    30.78 
89  36    30.83 
89.33    30.87 
89.29    30.92 
89.26    30.97 
89.22    31.01 

12 
14 

16 
18 
20 

92.22    26.79 
92.19    26.84 
92.15    26.89 
92.12    26.94 
92.09    26.99 

91.26    28.25 
91.22    28.30 
91.19    28.34 
91.16    28.39 
91.12    28.44 

90.24    29.67 
90.21    29.72 
90.18    29.76 
90.14    29.81 
90.11    29.86 

89.18    31.06 
89.15    31.10 
89  11     31.15 
89.08    31.19 
89.04    31.24 

22 
24 
26 
28 
30 

92.06    27.04 
92.03    27.09 
92.00    27.13 
91.97    27.18 
91.93    27.23 

91.09    28.49 
91.06    28.54 
91.02    28.58 
90.99    28.63 
90.96    28.68 

90.07    29.90 
90.04    29.95 
90.00    30.00 
89.97    30.04 
89.93    30.09 

89.00    31.28 
88.96    31.33 
88.93    31.38 
88.89    31.42 
88.86    31.47 

32 
34 
36 
38 
40 

91.90    27.28 
91.87    27.33 
91.84    27.38 
91.81    27.43 
91.77    27.48 

90.92    28.73 
90.89    28.77 
90.86    28.82 
90.82    28.87 
90.79    28.92 

89.90    30.14 
89.86    30.19 
89.83    30.23 
89.79    30.28 
89.76    30.32 

88.82    31.51 
88.78    31.56 
88.75    31.60 
88.71    31.65 
88.67    31.69 

42 
44 
46 

48 
50 

91.74    27.52 
91.71    27.57 
91.68    27.62 
91.65    27.67 
91.61    27.72 

90.76    28.96 
90.72    29.01 
90.69    29.06 
90.66    29.11 
90.62    29.15 

89.72    30.37 
89.69    30.41 
89.65    30.46 
89.61     30.51 
89.58    30.55 

88.64    31.74 
88.60    31.78 
88.56    31.83 
88.53    31.87 
88.49    31.92 

52 
54 
56 
58 
60 

91.58    27.77 
91.55    27.81 
91.52    27.86 
91.48    27.91 
91.45    27.96 

90.59    29.20 
90.55    29.25 
90.52    29.30 
90.48    29.34 
90.45    29.39 

89.54    30.60 
89.51    30.65 
89.47    30.69 
89.44    30.74 
89.40    30.78 

88.45    31.96 
88.41    32.01 
88.38    32.05 
88.34    32.09 
88.30    32.14 

c+/=    .75 
c  +  /=1.00 
c+/=1.35 

.72        .21 
.96        .28 
1.30        .36 

.72        .23 
.95        .30 
1.19        .38 

.71        .24 
.95        .32 
1.19        .40 

.71        .25 
.94        .33 
1.18        .42 
' 

TABLE  X.      STADIA  REDUCTIONS. 


167 


TABLE  X. 

STADIA  REDUCTIONS  FOB  READING   100. 


Minutes. 

20° 

21° 

22° 

23° 

Hor.      Diff. 
Dist.     Elev. 

Hor.       Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

0' 
2 
4 
6 
8 
10 

88.30    32.14 
88.26    32.18 
88.23    32.23 
88.19    32.27 
88.15    32.32 
88.11    32.36 

87.16    33.46 
87.12    33.50 
87.08    33.54 
87.04    33.59 
87.00    33.63 
86.96    33.67 

85.97    34.73 
85.93    34.77 
85.89    34.82 
85.85    34.86 
85.80    34  90 
85.76    34.94 

84.73    35.97 
84.69    36.01 
84.65    36.05 
84.61    36.09 
84.57    36.13 
84.52    36.17 

12 
14 
16 
18 
20 

88.08    32.41 
88.04    32.45 
88.00    32.49 
87.96    32.54 
87.93    32.58 

86.92    33.72 
86.88    33.76 
86.84    33.80 
86.80    33.84 
86.77    33.89 

85.72    34.98 
85.68    35.02 
85.64    35.07 
85.60    35.11 
85.56    35.15 

84.48    36.21 
84.44    36.25 
84.40    36.29 
84.35    36.33 
84.31    36.37 

22 
24 

26 
28 
30 

87.89    32.63 
87.85    32.67 
87.81    32.72 
87.77    32.76 
87.74    32.80 

86.73    33.93 
86.69    33.97 
86.65    34.01 
86.61    34.06 
86.57    34.10 

85.52    35.19 
85.48    35.23 
85.44    35.27 
85.40    35.31 
85.36    35  36 

84.27    36.41 
84.23    36.45 
84.18    36.49 
84.14    36.53 
84.10    36.57 

32 
34 
36 
38 

40 

87.70    32.85 
87.66    32.89 
87.62    32.93 
87.58    32.98 
87.54    33.02 

86.53    34.14 
86.49    34.18 
86.45    34.23 
86.41    34.27 
86.37    34.31 

85.31    35.40 
85.27    35.44 
85.23    35.48 
85.19    35.52 
85.15    35.56 

.84.06    36'.  61 
84.01    36.65 
83.97    36.69 
83.93    36.73 
83.89    36.77 

42 
44 
46 
48 
50 

87.51    33.07 
87.47    33.11 
87.43    33.15 
87.39    33.20 
87.35    33.24 

86.33    34.35 
86.29    34.40 
86.25    34.44 
86  21    34.48 
86.17    34.52 

85.11    35.60 
85.07    35.64 
85.02    35.68 
84.98    35.72 
84.94    35.76 

83.84    36.80 
83.80    36.84 
83.76    36.88 
83.72    36.92 
83.67    36.96 

52 
54 
56 

58 
60 

87.31    33.28 
87.27    33.33 
87.24    33.37 
87.20    33.41 
87.16    33.46 

86.13    34.57 
86.09    34.61 
86.05    34.65 
86.01     34.69 
85.97    34.73 

84.90    35.80 
84.86    35.85 
84.82    35.89 
84.77    35.93 
84.73    35.97 

83.63    37.00 
83.59    37.04 
83.54    37.08 
83.50    37.1-2 
83.46    37.16 

c+/=    -75 
c+/=1.00 
c-f  /=  1.25 

.70        .26 
.94        .35 
1.17        .44 

.70        .27 
.93        .37 

1.16        .46 

.69        .29 
.92        .38 
1.15        .48 

.69        .30 
.92        .40 
1.15        .50 

168 


TABLE  X.      STADIA   REDUCTIONS. 


TABLE  X. 

STADIA  REDUCTIONS   FOR  READING   100. 


Minutes. 

24° 

25° 

26° 

27° 

Hor.      Diff. 
Dist.     Elev. 

Hor.      Diff. 
Dist.     Elev. 

Hor.       Diff. 
Dist.     Elev. 

Hor.       Diff. 
Dist.     Elev. 

0' 
2 
4 
6 

8 

;o 

83.46    37.16 
83.41    37.20 
8337    37.23 
83.33    37.27 
83.28    37.31 
83.24    37.35 

82.14    38.30 
82.09    38.34 
82.05    38.38 
82.01    38.41 
81.96    38.45 
81.92    38.49 

80.78    39.40 
80.74    39.44 
80.69    39.47 
80.65    39.51 
80.60    39.54 
80.55    39.58 

79.39    40.45 
79.34    40  49 
79.30    40.52 
79.25    40.55 
79  20    40.59 
79.15    40.62 

12 

14 
16 
18 
20 

83.20    37.39 
83.15    37.43 
83.11     37.47 
83  07    37.51 
83.02    37.54 

81.87    38.53 
81  83    38.56 
81  .78    38  60 
81.74    38.64 
81.69    38.67 

80.51    39.61 
80.46    39.65 
80.41    39.69 
80.37    39.72 
80.32    39.76 

79.11     40.66 
79.06    40.69 
79.01    40.72 
78.96    40.76 
78.92    40.79 

22 
24 
26 

28 
30 

82.98    37.58 
82.93    37.62 
82.89    37.66 
82.85    37.70 
82.80    37.74 

81.65    38.71 
81.60    38.75 
81.56    38.78 
81.51    38.82 
81.47    38.86 

80.28    39.79 
80.23    39.83 
80.18    39.86 
80.14    39.90 
80.09    39.93 

78.87    40  82 
78.82    40.86 
78.77    40  89 
78.73    40.92 
78.68    40.96 

32 
34 
36 
38 
40 

82.76    37.77 
82.72    37.81 
82.67    37.85 
82.63    37.89 
82.58    37.93 

81.42    38.89 
81.38    38.93 
81.33    38.97 
81.28    39.00 
81.24    39.04 

80.04    39.97 
80.00    40.00 
79.95    40.04 
79.90    40.07 
79.86    40.11 

78.63    40.99 
78.58    41.02 
78.54    41.06 
78.49    41.09 
78.44    41.12 

42 
44 
46 
48 

f)0 

82.54    37.96 
82.49    38.00 
82.45    38.04 
82.41    38.08 
82.36    38.11 

81.19    39.08 
81.15    39.11 
81.10    39.15 
81.06    39.18 
81.01    39.22 

79.81    40.14 
79.76    40.18 
79.72    40.21 
79.67    40.24 
79.62    40.28 

78.39    41.16 
78.34    41.19 
78.30    41.22 
78.25    41.26 
78.20    41.29 

52 
54 
56 

58 
60 

82.32    38.15 
82.27    38.19 
82.23    38.23 
82.18    38.26 
82.14    38.30 

80.97    39.26 
80.92    39.29 
80.87    39.33 
80.83    39.36 
80.78    39.40 

79.58    40.31 
79.53    40.35 
79.48    40.38 
79.44    40.42 
79.39    40.45 

78.15    41.32 
78.10    41.35 
78.06    41.39 
78.01     41.42 
77.96    41.45 

c+/=    .75 
c+f=  1.00 
c+/=1.25 

.68        .31 
.91         .41 
1.14        .52 

.68        .32 
.90        .43 
1.13        .54 

.67        .33 
.89        .45 
1.12        .56 

.66        .35 
.89        .46 
1.11         .58 

TABLE  XL 
LOGARITHMS   OF  NUMBERS 

FROM 

1  to  10  000 

TO  SIX  DECIMAL  PLACES. 


N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

I 
|  N. 

Log. 

1 

0.000000 

21 

1.322219 

41 

1.612784 

61 

1.785330 

81 

1.908485 

2 

0.301030 

22 

1.342423 

42 

1.623249 

62 

1.792392 

1  82 

1.913814 

3 

0.477121 

23 

1.361728 

43 

1.633468 

63 

1.799341 

83 

.919078 

4 

0.602060 

24 

1.380211 

44 

1.643153 

64 

.806180 

84 

.924279 

5 

0.698970 

25 

1.397940 

45 

1.653213 

65 

.812913 

85 

.929419 

6 

0.778151 

26 

1.414973 

46 

1.662758 

66 

.819544 

86 

.934498 

0.845098 

27 

1.431364 

47 

1.672098 

67 

.826075 

87 

.939519 

8 

0.903090 

28 

1.447158 

48 

1.681241 

68 

.832509 

88 

.944483 

9 

0.954243 

29 

1.462398 

49 

1.690196 

69 

.838849 

89 

.949390 

10 

1.000000 

30 

1.477121 

50 

1.698970 

70 

.&45098 

90 

.954243 

11 

1.041393 

31 

1.491362 

51 

.707570 

71 

.851258 

91 

.959041 

12 

1.079181 

32 

1.505150 

52 

.716003 

72 

.857332 

92 

.963788 

13 

1.113943 

33 

1.518514 

53 

.724276 

73 

.863323 

93 

.968483 

14 

1.146128 

34 

1.531479 

54 

.732394 

74 

.869232 

94 

.973128 

15 

1.176091 

35 

1.544068 

55 

.740363 

75 

.875061 

95 

.977724 

16 

1  204120 

36 

1.556303 

56 

.748188 

76 

.880814 

96 

1.982271 

17 

1.230449 

37 

1.568202 

57 

.755875 

77 

.886491 

97 

1.986772 

18 

1.255273 

38 

1.579784  1 

58 

1.763428 

78 

.892095 

98 

1.991226 

19 

1.278754 

39 

1.591065 

59 

1.770852 

79 

.897627 

99 

1.995635 

SO 

1.301030 

40 

1.602060 

60 

1.778151 

80 

.903090 

100 

2.000000 

169 


TABLE  XI.      LOGARITHMS  OF  NUMBERS. 


No. 

100  L.  000.] 

[No.  109  L.  040. 

N. 

0 

1          2 

8         4 

5 

6         7 

8          9 

Diff. 

100 

000000 

0434     0868 

1301     1734 

2166 

2598     3029 

3461      3891 

432 

1 

4321 

4751      5181 

5609     6038 

6466 

6894     7321 

7748     8174 

426 

2 

8600 

9026     9451 

9876 

0300 

0724 

1147     1570 

1993     2415 

AOA 

3 

012837 

3259     3680 

4100     4521 

4940 

5360  \  5779 

6197     6616 

WDI 

420 

7033 

7451     7868 

8284     8700 

9116 

9532     9947 

1 

0361     0775 

...  _ 

5 

021189 

1603     2016 

2428     2841 

3252 

3664     4075 

4486     4896 

412 

6 

5306 

5715     6125 

6533     6942 

7350 

7757     8164 

8571     8978 

408 

7 

QOQJ. 

9^89   

I 

yoot 

0195 

0600     1004 

1408 

1812     2216 

2619     3021 

.. 

8 

033424 

3826     4227 

4628     5029 

5430 

5830     6230 

6629     7028 

400 

7426 

7825     8223 

8620     9017 

98ll 

04 

0207 

0602     0998 

397 

PROPORTIONAL  FAHTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

6 

9 

434 

43.4 

86.8 

130.2 

173,6 

217.0       260.4 

3( 

)3.8 

347.2 

390.6 

433 

43.3 

86.6 

129.9 

173.2 

216.5 

259.8 

K 

)3.1 

346.4 

389.7 

432 

43.2 

86.4 

129.6 

172.8 

216.0 

259.2 

ft 

)2.4 

345.6 

388.8 

431 

43.1 

86.2 

12 

9.3 

172.4 

215.5 

258.6 

3( 

)1.7 

344.8 

387.9 

430 

43.0 

86.0 

129.0 

172.0 

215.0 

258.0 

301.0 

344.0 

387.0 

429 

42.9 

85.8 

128.7 

171.6 

214.5 

257.4 

300.3 

343.2 

386.1 

428 

42.8 

85.6 

128.4 

171.2 

214.0 

256.8 

299.6 

342.4 

385.2 

427 

42.7 

85.4 

12 

8.1 

170.8 

213.5 

256.2 

2< 

)8.9 

341.6 

384.3 

426 

42.6 

85.2 

127.8 

170.4 

213.0 

255.6 

298.2 

340.8 

383.4 

425 

42.5 

85.0 

127.5 

170.0 

212.5 

255.0 

297.5 

340.0 

382.5 

424 

42.4 

84.8 

127.2 

169.6 

212.0 

254.4 

2< 

)6.8 

339.2 

381.6 

423 

42.3 

84.6 

12 

6.9 

169.2 

211.5 

253.8 

2< 

)6.1 

338.4 

380.7 

422 

42.2 

84.4 

126.6 

168.8 

211.0 

253.2 

295.4 

337.6 

379.8 

421 

42.1 

84.2 

126.3 

168.4 

210.5 

252.6 

294.7 

336.8 

378.9 

420 

42.0 

84.0 

126.0 

168.0 

210.0 

252.0 

2" 

M.O 

"336.0 

378.0 

419 

41.9 

83.8 

12 

5.7 

167.6 

209.5 

251.4 

2< 

)3.3 

335.2 

377.1 

418 

41.8 

83.6 

125.4 

167.2 

209.0 

250.8 

292.6 

334.4 

376.2 

417 

41.7 

83.4 

12 

5.1 

166.8 

208.5 

250.2 

2« 

41.9 

333.6 

375.3 

416 

41.6 

83.2 

124.8 

166.4 

208.0 

249.6 

291.2 

332.8 

374.4 

415       41.5 

83.0 

124.5 

166.0 

207.5 

249.0 

290.5 

332.0 

373.5 

414 

41.4 

82.8 

124.2 

165.6 

207.0 

248.4 

289.8 

331.2 

372.6 

413 

41.3 

82.6 

12 

165.2 

206.5 

247.8 

2 

39.1 

330.4 

371.7 

412 

41.2 

82.4 

123.6 

164.8 

206.0 

247.2 

288.4 

329.6 

370.8 

411 

41.1 

82.2 

123.3 

164.4 

205.5 

246.6 

287.7 

328.8 

369.9 

410 

4UO 

82.0 

IS 

5.0 

164.0 

205.0 

246.0 

2 

37.0 

328.0 

369.0 

409 

40.9 

81.8         122.7 

163.6 

204.5 

245.4 

2 

36.3 

327.2 

368.1 

408 

40.8 

81.6    1    122.4 

163.2 

204.0 

244.8 

285.6 

326.4     367.2 

407 

40.7 

81.4 

IS 

»2.1 

162.8 

203.5 

244.2 

2 

84.9 

325.6  j  366.3 

406 

40.6 

81.2 

121.8 

162.4 

208.0 

243  6 

284.2 

324.8     365.4 

405 

40.5 

81.0 

121.5 

162.0 

202.5 

243.0 

2 

33.5 

324.0     364.5 

404 

40.4 

80.8 

-  121.2 

161.6 

202.0 

242.4 

2 

82.8 

323.2     363.6 

403 

40.3 

80.6 

120.9 

161.2 

201.5 

241.8 

2 

82.1 

322.4     362.7 

40$ 

! 

40.2 

80.4 

1$ 

160.8 

201.0 

241  2 

2 

81.4 

321.6     361.8 

401 

40.1 

80.2 

120.3 

160.4 

200.5 

240.6 

280.7 

320.8     360.9 

4(K 

) 

40.0 

80-0 

li 

JO.O 

160.0 

200.0 

240.0 

3 

80.0 

320.0 

360.0 

399 

39.9 

79.8 

119.7 

159.6 

199.5 

239.4 

279.3 

319.2 

359.1 

39* 

1 

39.8 

79.6 

1 

19.  4 

159.2 

199.0 

238.8 

2 

78.6 

318.4 

358.2 

397 

39.7 

79.4 

119.1 

158.8 

198.5 

238.2 

277.9 

317.6 

357.3 

39( 

3 

39.6 

79.2 

1 

18.8 

158.4 

198.0 

237.6 

2 

77.2 

316.8 

356.4 

395 

39.5 

79.0         118.5 

158.0 

197.5       237.0       276.5       316.0     355.5 

TABLE  XI.      LOGARITHMS  OF  NUMBERS. 


No. 

110  L.  041.] 

[No.  119  L.  078. 

N. 

0 

1          2 

3         4 

5 

6 

7          8 

9 

Diff. 

110 

041393 

1787     2182 

2576     2969 

3362 

3755 

4148     4540 

4932 

393 

1 

5323 

5714     6105 

6495     6885 

7275     7664 

8053     8442 

8830 

390 

2 

9218 

9606     9993 

0380     0766 

1153 

1KQQ 

1  Q94       9^ftQ 

9fiQJ. 

qoc 

3 

053078 

3463     3846 

4230     4613 

4996 

1OOO 

5378 

5760     6142 

6524 

OoD 

383 

4 

6905 

7286     7666 

8046     8426 

8805 

9185 

9563     9942 

0320 

379 

5 

060698 

1075      1452 

1829     2206 

2582 

2958 

3333     3709 

4083 

376 

6 

4458 

4832     5206 

5580     5953 

6326 

6699 

7071     7443 

7815 

373 

7 

8186 

8557     8928 

9298     9668 

0038 

0407 

0776      1145 

1514 

370 

8 

071882 

2250     2617 

2985     3352 

3718 

4085 

4451      4816 

5182 

366 

9 

5547 

5912     6276 

6640     7004 

7368 

7731 

8094     8457 

8819 

363 

PROPORTIONAL  PARTS. 

Diff. 

1 

1 

3 

4 

5 

6 

7 

8 

9 

395 
394 

39.5 
39.4 

79.0 

78.8 

118.5 
118.2 

158.0 
157.6 

197.5 
197.0 

237 
236 

.0 
.4 

276.5 
275.8 

316.0 
315.2 

355.5 
354.6 

393 

39.3 

78.6 

117.9 

157.2       196.5 

235 

.8 

275.1 

314.4 

353.7 

392 

39.2 

78.4 

117.6 

156.8       196.0 

235 

2 

27  4  A 

313.6 

352.8 

391 

39.1 

78.2 

117.3 

156.4       195.5 

234 

'.6 

273.7 

312.8 

351.9 

39C 

39.0 

78.0 

117.0 

156.0 

195.0 

234 

.0 

273.0 

312.0 

351.0 

38S 

33.9 

77.8 

116.7 

155.6 

194.5 

233.4 

272.3 

311.2 

350.1 

38£ 

38.8 

77.6 

116.4 

155.2 

194.0 

232 

.8 

271.6 

310.4 

349.2 

387 

38.7 

77.4 

116.1 

154.8 

193.5 

232.2 

270.9 

309.6 

348.3 

386 

38.6 

77.2 

115.8 

154.4 

193.0 

231 

.6 

270.2 

308.8 

347.4 

38E 

38.5 

77.0 

115.5 

154.0 

192.5 

231 

.0 

269.5 

308.0 

346.5 

384 

38.4 

76.8 

115.2 

153.6 

192.0 

230.4 

268.8 

307.2 

"345.6 

38? 

1 

38.3 

76.6 

114.9 

153.2 

191.5 

22S 

.8 

268.1 

306.4 

344.7 

38$ 

1 

38.2 

76.4 

114.6 

152,8 

191.0 

22£ 

.2 

267.4 

305.6 

343.8 

381 

38.1 

76.2 

114.3 

152.4 

190.5 

22£ 

.6 

266.7 

304.8 

342.9 

380 

38.0 

76.0 

114.0 

152.0 

190.0 

22£ 

.0 

266.0 

304.0 

342.0 

371 

1 

37.9 

75.8 

113.7 

151.6 

189.5 

227 

'.4 

265.3 

303.2 

341.1 

378 

37.8 

75.6 

113.4 

151.2 

189.0 

226.8 

264.6 

302.4 

340.2 

37' 

r 

37.7 

75.4 

113.1 

150.8 

188.5 

22( 

>.2 

263.9 

301.6 

339.3 

376 

37.6 

75.2 

112.8 

150.4 

188.0 

2£ 

263.2 

300.8 

338.4 

375 

37.5 

75.0 

112.5 

150.0 

187.5 

225.0 

262.5 

300.0 

337.5 

374 

37.4 

74.8 

112.2 

149.6 

187.0 

224 

L4 

261.8 

299.2 

336.6 

37 

3 

37.3 

74.6 

111.9 

149.2 

186.5 

22C 

5.8 

261.1 

298.4 

335.7 

372 

37.2 

74.4 

111.6 

148.8 

186.0 

223.2 

260.4 

297.6 

334.8 

37 

i 

37.1 

74.2 

111.3 

148.4 

185.5 

22$ 

J.6 

259.7 

296.8 

333.9 

370 

37.0 

74.0 

111.0 

148.0 

185.0 

222.0 

259.0 

296.0 

333.0 

36 

i 

36.9 

73.8 

110.7 

147.6 

184.5 

221 

.4 

258.3 

295.2 

332.1 

361 

3 

36.8 

73.6 

110.4 

147.2 

184.0 

220.8 

257.6 

294.4 

331.2 

36 

i 

36.7 

73.4 

110.1 

146.8 

183.5 

22( 

256.9 

293.6 

830.3 

36 

5 

36.6 

73.2 

109.8 

146.4 

183.0 

211 

K6 

256.2 

292.8 

329.4 

S65 

36.5 

73.0 

109.5 

146.0 

182.5 

219.0 

255.7 

292.0 

328.5 

364 

36.4 

72.8 

109.2 

145.6 

182.0 

218.4 

254.8 

291.2 

327.6 

36 

3 

36.3 

72.6 

108.9 

145.2 

181.5 

21r 

l'S 

254.1 

290.4 

326.7 

36 

2 

36.2 

72.4 

108.6 

144.8 

181.0 

21r 

253.4 

289.6 

325.8 

361 

36.1 

72.2 

108.3 

144.4 

180.5 

216.6 

252.7 

•  288.8 

324.9 

36 

0 

36.0 

72.0 

108.0 

144.0 

180.0 

216.0 

252.0 

288.0 

324.0 

35 

9 

35.9 

71.8 

107.7 

143.6 

179.5 

2tt 

251.3 

287.2 

323.1 

35 

8 

35.8 

71.6 

107.4 

143.2 

179.0 

214 

L8 

250.6 

286.4 

322.2 

35 

7 

35  7 

71.4 

107.1 

142.8 

178.5 

214 

L2 

249.9 

285.6 

321.3 

356 

35.6 

71.2 

106.8 

142.4 

178.0 

213.6 

249.2 

284.8 

320.4 

172 


TABLE   XI.      LOGARITHMb   OF   LUMBERS. 


I  No.  120  L.  079.] 


[No.  134  L.  130. 


N. 

0 

1 

2 

3 

4  j|  6 

6 

7 

8 

9 

Diff. 

120 

079181 

9543 

9904 

i 

0266  0626  !l  0987 

1347 

1707 

2067  2426 

360 

1 

082785 

3144 

a503 

3861  4219 

4576 

4934 

5291 

5647 

6004 

357 

2 

6360 
9905 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

355 

. 

0258 

0611 

0963 

1315 

1667 

2018 

2370 

2721 

3071 

352 

4 

093422 

3772 

4122 

4471 

4820 

5169 

5518 

5866 

6215 

6562 

349 

5 

6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

0026 

346 

6 

100371 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

343 

7 

3804 

4146 

4487 

4828 

5169 

5510 

5851 

6191 

6531 

6871 

341 

8 

7210 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

0253 

338 

9  110590 

0926 

1263 

1599 

1934 

i  2270 

2605 

2940 

3275 

3609 

335 

130    3943 

4277 

4611 

4944 

5278 

I  5611 

5943 

6276 

6608 

6940 

333 

7971 

7603 

7934 

8265 

8595 

8926 

9256 

9586 

9915 

0245 

330 

2  120574 

0903 

1231 

1560 

1888 

2216 

2544 

2871 

3198 

3525 

328 

3 

3852 

4178 

4504 

4830 

5156 

5481 

5806 

6131 

6456 

6781 

325 

4 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

13 

0012    3213 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

355 

35.5 

71.0 

106.5 

142.0 

177.5   213.0 

248.5 

284.0 

319.5 

354 

35.4 

70.8 

106.2 

141.6 

177.0  i  212.4 

247.8 

283.2 

318.6 

353 

35.3 

70.6 

105.9 

141.2 

176.5  1  211.8 

247.1 

282.4 

317.7 

352 

35.2 

70.4 

105 

.6 

140.8 

176.0 

211.2 

246.4 

,281.6 

316.8 

351 

35.1 

70.2 

105.3 

140.4 

175.5 

210.6 

245.7 

280.8 

315.9 

350 

35.0 

70.0 

105.0 

140.0 

175.0 

210.0 

245.0 

280.0 

315.0 

349 

34.9 

69.8 

104 

.7 

139.6 

174.5 

209.4 

244.3 

279.2 

314.1 

348 

34.8 

69.6 

104.4 

139.2 

174.0 

208.8 

243.6 

278.4 

313.2 

347 

34.7 

69.4 

104 

.1 

138.8 

173.5 

208.2 

242.9 

277.6 

312.3 

346 

34.6 

69.2 

103.8 

138.4 

173.0 

207.6 

242.2 

276.8 

311.4 

345 

34.5 

69.0 

103 

.5 

138.0 

172.5 

207.0 

241.5 

276.0 

310.5 

344 

34.4 

68.8 

103 

.2 

137.6 

172.0 

20G.4 

240.8 

275.2 

309.6 

343 

34.3 

68.6 

102 

.9 

137.2 

171.5 

205.8 

240.1 

274.4 

308.7 

ai2 

34.2 

68.4 

102 

.6 

136.8 

171.0 

205  2 

239.4 

273.6 

307.8 

341 

34.1 

68.2 

102 

.3 

136.4 

170.5 

204.6 

238.7 

2V2.8 

306.9 

340 

34.0 

68.0 

102 

.0 

136.0 

170.0 

204.0 

238.0 

272.0 

306.0 

339 

33.9 

67.8 

101.7 

135.6 

169.5 

203.4 

237.3 

271.2 

305.1 

838 

33.8 

67.6 

101 

.4 

135.2 

169.0 

202.8 

236.6 

270.4 

304.2 

337 

33.7 

67.4 

101 

.1 

134.8 

168.5 

202.2 

235.9 

269.6 

303.3 

336 

33.6 

67.2 

100 

.8 

134.4 

168.0 

201.6 

235.2 

2G8.8 

302.4 

335 

33.5 

67.0 

100 

.5 

134.0 

167.5 

201.0 

234.5 

268.0 

301.5 

334 

33.4 

66.8 

100 

.2 

133.6 

167.0 

200.4 

233  8 

267.2 

300.6 

333 

as.  3 

66.6 

99.9 

133.2 

166.5 

199.8 

233.1 

266.4 

299.7 

332 

33.2 

66.4 

99 

.6 

132.8 

166.0 

199.2 

232.4 

265.6 

298.8 

331 

33.1 

66.2 

99.3 

132.4 

165.5 

198.6  i  231.7 

264.8 

297.9 

330 

as.o 

66.0 

99 

.0 

132.0 

165.0 

198.0 

231.0 

264.0 

297.0 

329 

32.9   65.8 

98 

.7 

131.6 

164.5 

197.4 

230.3 

263.2 

296.1 

328 

32.8   65.6 

98.4 

131.2 

164.0   196.8 

229.6 

262.4 

295.2 

327 

32.7 

65.4 

98 

.1 

130.8 

163.5   196.2 

228.9 

261.6 

294.3 

326 

32.6 

65.2 

97 

.8 

130.4 

163.0   195.6 

228.2 

260.8 

293.4 

325 

32.5 

65.0 

97 

.5 

130.0 

162.5 

195.0 

227.5 

260.0 

292.5 

324 

32.4 

64.8 

97 

.2 

129.6 

162.0 

194.4 

226.8 

259.2 

291.6 

323 

32.3 

64.6 

96 

.9 

129.2 

161.5 

193.8 

226.1 

258.4 

290.7 

322  I  32  2 

64.4 

96 

.6 

128.8   161.0 

193.2 

225.4 

257.6 

289.8 

TABLE   XT.      LOGARITHMS   OF    NUMBERS. 


173 


No.  135  L.  130.] 

[No.  149  L.  175. 

N.          0 

1 

2 

3 

4 

5         6 

7 

8 

9 

Diff. 

135     130334 

0655 

0977 

1298 

1619       1939     2260 

2580 

2900 

3219 

321 

6         3539 

3858 

4177 

4496 

4814      5133     5451 

5769 

6086 

6403 

318 

7         6721 
8        9879 

7037 

7354 

7671 

7987      8303  j  8618 

8934 

9249 

9564 

316 

0194 

0508 

0822 

1136 

1450     1763 

2076 

2389 

2702 

314 

9     143015 

3327 

3639 

3951 

4263 

4574 

4885 

5196 

5507 

5818 

311 

140         6128 

^ 

6748 

9835 

7'058 

7367 

7676     7985 

8294 

8603 

8911 

309 

i        y^iy 

0142 

0449 

0756     i<~»AQ 

1370 

1676 

1982 

307 

2     152288 

2594 

2900 

3205 

3510 

3815 

4120 

4424 

4728 

5032 

305 

3         5336 

5640 

ocrM 

5943 

QfUJK 

6246 

92G6 

6549 
9567 

6852 
9868 

7154 

7457 

7759 

8061 

303 

4 

OOO't 

oyoo 

0168 

0469 

0769 

1068 

301 

5     131368 

1667 

1967 

2266 

2564 

2863     3161 

3460 

3758 

4055 

299 

6         4353 

4650 

4947 

5244 

5541 

5838 

6134 

6430 

6726 

7022 

297 

7         7317 

7613 

7908 

8203 

8497 

8792  ;  9086 

9380 

9674 

9968 

295 

8     170262 

0555 

0848 

1141 

1434 

1726 

2019 

2311 

2603 

2895 

293 

9         3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

5512 

5802 

291 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

821 

32.1 

64.2 

96 

.3 

128.4 

160.5 

192 

0 

224.7 

256.8 

288.9 

320 

32.0 

64.0 

96.0 

128.0 

160.0 

192.0 

224.0 

256.0 

288.0 

319 

31.9 

63.8 

95 

.7 

127.6 

159.5 

191 

4 

£ 

!3.3 

255.2 

287.1 

318 

31.8 

63.6 

95 

.4 

127.2 

159  0 

190 

8 

222.6 

254.4 

286.2 

317 

31.7 

63.4 

95 

.1 

126.8 

158.5 

190 

2 

$ 

11.9 

253.6 

285.3 

316 

31.6 

63.2 

94 

.8 

126.4 

158.0 

189.6 

221.2  ! 

252.8 

284.4 

315 

31.5 

63.0 

94 

.5 

126.0 

157.5 

189 

0 

25 

JO.  5 

252.0 

283.5 

314 

31.4 

62.8 

94 

.2 

125.6 

157.0 

188 

4 

21 

9.8  ! 

251.2 

282.6 

313 

31.3       62.6 

93 

.9 

125.2 

156.5 

187 

6 

219.1  ! 

250.4 

281.7 

312 

31.2  :     62.4 

93.6 

124.8 

156.0 

187 

2 

218.4  1 

249.6 

280.8 

311 

31.1 

62.2 

93 

.3 

124.4 

155.5 

186 

6 

21 

7.7 

248.8 

279.9 

310 

31.0       62.0 

93 

.0 

124.0 

155.0 

186 

0 

217.0  ! 

248.0 

27'9.0 

309 

30.9  !     61.8 

92 

7 

123.6 

154.5 

185.4 

216.3 

247.2     278.1 

308 

30.8       61.6         92 

'A 

123.2 

154.0        184 

8 

21 

5.6 

246.4     277.2 

307 

30.7       61.4 

92.1 

122.8 

153.5       184.2 

214.9 

245.6     27'6.3 

306 

30.6       61.2         91 

.8 

122.4 

153,0       183 

6       2 

14.2 

244.8 

275.4 

305 

30.5       61.0         91 

.5 

122.0 

152.5       183 

0       21 

L3.5 

244.0     274,5 

304 

30.4       60.8    i     91 

.2 

121.6 

152.0  i     182.4 

212.8 

243.2 

273.6 

303 

30.3       60.6     i     90 

.9 

121.2 

151.5 

181 

8      -21 

12.1 

°/42.4 

272.7 

302 

30.2       60.4 

90 

.6 

120.8 

151.0 

181 

2 

211.4 

241.6     271.8 

301 

30.1       60.2 

90.3 

120  4 

150.5 

180.6 

210.7 

240.8     270.9 

300 

30.0       60.0 

90 

.0 

120.0 

150.0 

180 

0 

2 

0.0 

240.0 

270.0 

299 

29.9       59.8 

89.7 

119.6 

149.5 

179 

4       209.3 

239.2 

269.1 

298 

29.8       59.6 

89 

.4 

119.2 

149.0 

178 

8 

2( 

)8.6  i 

238.4     268.2 

297 

29.7       59.4 

89.1 

118.8 

148.5 

178 

2 

207.9 

237.6     267.3 

296 

29.6       59.2         88 

.8 

118.4 

148.0 

177 

6 

2( 

)7.2 

236.8     266.4 

295 

29.5       59.0         88.5 

118.0 

147.5 

177 

0 

206.5 

236.0     265.5 

294 

29.4       58.8 

88 

.2 

117.6 

147.0 

176 

4       2( 

)5.8 

235.2     264.6 

293 

29.3 

58.6 

87 

.9 

117.2 

146.5 

175 

8 

205.1 

234.4     263.7 

292 

29.2 

58.4 

87 

.6 

116.8 

146.0 

175.2 

2 

'4.4 

233.6     262.8 

291 

29.1 

58.2        87.3^ 

116.4 

145.5 

174 

6 

a 

)3.7 

232.8  ;  261.9 

290 

29.0 

58.0    j    87 

.0 

116.0 

145.0 

174.0 

203.0   : 

232.0  ,  2(51.0 

289 

28.9 

57.8         86 

.7 

115.6 

144.5       173 

4 

2( 

)2.3  1 

231.2  i  260.1 

288 

28.8 

57.6         86 

.4 

115.2 

144.0 

172 

8 

to 

)1.6 

230.4  i  259.2 

287 

28.7 

57.4        86 

.1 

114.8 

143.5 

172 

2 

2( 

X).9 

229.6     258.3 

286 

28.6 

57.2        85 

.8 

114.4 

143.0 

171.6 

200.2 

228.8     257.4 

174 


TABLE  XI.      LOGARITHMS   OF  NUMBERS, 


No.  150  L.  176.] 

[No.  169  L.  230. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

150  176091 

6381 

6G70 

6959 

7248 

7536 

7825 

8113 

8401 

8689 

289 

QQ77 

9264 

9552 

9839 

aal  t 

0126 

0413 

flfqq   nuxr, 

1272 

1558 

287 

2 

181844  2129 

2415 

2700 

2985 

3270 

3555 

3839 

4123 

4407 

285 

3 

4691 

4975 

5259 

5542 

5825 

6108 

6391 

6674 

6956  7239 

283 

7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

0051 

281 

5 

190332  0612 

0892 

1171 

1451  1  1730 

2010 

2289 

2567 

2846 

279 

6 

3125  3403 

3681 

3959  !  4237   4514 

4792 

5069 

5346 

5623 

278 

7 

5900  i  6176 

6453 

6729 

7005 

7281 

7556 

7832 

8107 

8382 

276 

9206 

9481 

9755 

ooo  i   oyorf 

0029 

0303 

(W"7 

0850 

1124 

274 

9 

201397 

1670 

1943 

2216 

2488 

2761 

3033 

3305 

3577 

3848 

272 

160 

4120 

4391 

4663 

4934 

5204 

5475 

5746 

6016 

6286 

6556 

271 

1 

6826  7096 

7365 

7634 

7904 

8173 

8441 

8710 

8979 

9247 

269 

2 

9515  9783 

0051 

0319 

0586 

0853 

1121 

1388 

1654 

1921 

267 

3 

212188 

2454 

2720 

2986 

3252 

3518 

3783 

4049 

4314 

4579 

266 

4 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

264 

5 

7484  7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

262 

6 

220108  0370 

0631 

0892 

1153 

1414 

1675 

1936 

2196 

2456 

261 

7 

2716 

2976 

3236 

3496 

3755 

4015 

4274 

4533 

4792 

5051 

259 

8 

5309  5568 

5826 

6084 

6342 

6600 

6858 

7115 

7372 

7630 

258 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

23 

0193 

256 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

285 

28.5 

57.0 

85.5 

114.0 

142.5   171.0 

199.5 

228.0 

256.5 

284 

28.4 

56.8 

85.2 

113.6 

142.0 

170.4 

198.8 

227.2  255.6 

283 

28.3 

56.6 

84 

.9 

113.2 

141.5 

169.8 

198.1 

226.4  254.7 

282 

28.2 

56.4 

84 

.6 

112.8 

141.0 

169.2 

197.4 

225.6  253.8 

281 

28.1 

56.2 

84.3 

112  4 

140.5 

168.6 

196.7 

224.8  252.9 

280 

28.0 

56.  C 

84 

.0 

112.0 

140.0 

168.0 

196.0   224.0  252.0 

279 

27.9 

55.8 

83.7 

111.6 

139.5 

167.4 

195.3   223.2  251.1 

278 

27.8 

55.  e 

83.4 

111.2 

139.0 

166.8 

194.6   222.4  250.2 

277 

27.7 

55.4 

83 

.1 

110.8 

138.5 

166.2 

193.9  !  221.6  !  249.3 

276 

27.6 

55.2 

82.8 

110.4 

138.0 

165.6 

193.2   220.8  j  248.4 

275 

27.5 

55.0 

82.5 

110.0 

137.5 

165.0 

192.5 

220.0 

247.5 

274 

27.4 

54.8 

82 

.2 

109.6 

137.0 

164.4  1  191.8   219.2  246.6 

273 

27.3 

54.6 

81 

.9 

109.2 

136.5 

163.8   191.1   218.4  245.7 

272 

27.2 

54.4 

81 

.6 

108.8 

136.0 

163.2   190.4  !  217.6 

244.8 

271 

27.1 

54.  S 

81 

.3 

108.4 

135.5 

162.6   189.7   216.8 

243.9 

270 

27.0 

54.0 

81.0 

108.0 

135.0 

162.0 

189.0   216.0 

243.0 

269 

26.9 

53.  6 

\ 

8C 

.7 

107.6 

134.5 

161.4 

188.3   215.2 

242.1 

268 

26.8 

53.6 

8C 

.4 

107.2 

134.0 

160.8 

187.6  !  214.4 

241.2 

267 

26.7 

63.4 

[ 

8C 

.1 

106.8 

133.5 

160.2 

186.9   213.6 

240.3 

266 

26.6 

53.2 

79.8 

106.4 

133.0 

159.6. 

186.2   212.8 

239.4 

265 

26.5 

53.0 

79.5 

106.0 

132.5 

159.0 

185.5   212.0 

238.5 

264 

26.4 

52.  £ 

] 

7£ 

.2 

105.6 

132.0 

158.4 

184.8  ,  211.2 

237.6 

263 

26.3 

52.  ( 

> 

78.9 

105.2 

131.5 

157.8 

184.1  !  210.4 

236.7 

262 

26.2 

52.4 

78.6 

104.8 

131.0 

157.2 

183.4  ;  209.6 

235.8 

261 

26.1 

52.5 

3 

78 

104.4 

130.5 

156.6 

182.7  !  208.8 

234.9 

260 

26.0 

52.0 

78.0 

104.0 

130.0   156.0 

182.0  !  208.0 

234.0 

259 

25.9 

51.  £ 

$ 

T 

.7 

103.6 

129.5   155.4 

181.3   207.2 

233.1 

258 

25.8 

51.  ( 

) 

77.4 

103.2 

129.0   154.8 

180.6  i  206.4 

232.2 

257 

25.7 

51.  < 

t 

r 

M 

102.8 

128.5   154.2 

179.9   205.6 

231.3 

256 

25.6 

51.2 

76.8 

102.4 

128.0   153.6 

179.2  1  204.8 

230.4 

255 

25.5 

51.0 

76.5 

102.0 

1£7.5  I  153.0 

178.5  i  204.0 

229.1 

TABLE   XI.      LOGARITHMS  OF   NUMBERS. 


No.  170  L.  230.]                                  [No.  189  L.  878. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

170 

230449 

0704 

0960 

1215 

1470 

1724 

1979 

2234 

2488 

2742 

255 

1 

2996 

3250 

3504 

3757 

4011 

4264 

4517 

4770 

5023 

5276 

253 

2 

5528 

5781 

6033 

6285 

6537 

6789 

7041 

7292 

7544 

7795 

252 

g 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

0050 

0300 

250 

4 

240549 

0799 

1048 

1297 

1546 

1795 

2044 

2293 

2541 

2790 

249 

5 

3038 

3286 

3534 

3782 

4030 

4277 

4525 

4772 

5019 

5266 

248 

6 

5513 

5759 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

7 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

0176 

245 

8 

250420 

0664 

0908 

1151 

1395 

1638 

1881 

2125 

2368 

2610 

243 

9 

2853 

3096 

sass 

3580 

3822 

4064 

4306 

4548 

4790 

5031 

242 

180 

5273 

5514 

5755 

5996 

6237 

6477 

6718 

6958 

7198 

7439 

241 

1 

7679 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

2 

260071 

0310 

0548 

0787 

1025 

1263 

1501 

1739 

1976 

2214 

238 

3 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

237 

4 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

5 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

g 

9513 

9746 

9980 

0213 

0446 

0679 

0912 

1144 

1377 

1609 

233 

7 

271842 

2074 

2306 

2538 

2770 

3001 

3233 

3464 

3696 

3927 

232 

8 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

5772 

6002 

6232 

230 

9 

6462 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

8296 

8525 

229 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

255 

25.5 

51.0 

76.5 

102.0 

127.5 

153.0 

178.5 

204.0 

229.5 

254 

25.4 

50.8 

76.2 

101.6 

127.0 

152.4 

177.8 

203.2 

228.6 

253 

25.3 

50.6 

75.9 

101.2 

126.5 

151.8 

177.1 

202.4 

227.7 

258 

25.2 

50.4 

75.6 

100.8 

126.0 

151.2 

176.4 

201.6 

226.8 

261 

25.1 

50.2 

75.3 

100.4 

125.5 

150.6 

175.7 

200.8 

225.9 

250 

25  0 

50.6 

75.0 

100.0 

125.0 

150.0 

175.0 

200.0 

225.0 

249 

24.9 

49.8 

74.7 

99.6 

124.5 

149.4 

174.3 

199.2 

224.1 

248 

24.8 

49.6 

74.4 

99.2 

124.0 

148.8 

173.6 

198.4 

223.2 

247 

24.7 

49.4 

74.1 

98.8 

123.5 

148.2 

172.9 

197.6 

222.3 

246 

24.6 

49.2 

73.8 

98.4 

123.0 

147.6 

172.2 

196.8 

221  .4 

245 

24.5 

49.0 

73.5 

98.0 

122.5 

147.0 

171.5 

196.0 

220.5 

244 

24.4 

48.8 

73.2 

97.6 

122.0 

146.4 

170.8 

195.2 

219.6 

243 

24.3 

48.6 

72.9 

97.2 

121.5 

145.8 

170.1 

194.4 

218.7 

242 

24.2 

48.4 

72.6 

96.8 

121.0 

145.2 

169.4 

193.6 

217.8 

241 

24.1 

48.2 

72.3 

96.4 

120.5 

144.6 

168.7 

192.8 

216.9 

240 

24.0 

48.0 

72.0 

96.0 

120.0 

144.0 

168.0 

192.0 

216.0 

239 

23.9 

47.8 

71.7 

95.6 

119.5 

143.4 

167.3 

191.2 

215.1 

238 

23.8 

47.6 

71.4 

95.2 

119.0 

142.8 

166.6 

190.4 

214.2 

237 

23.7 

47.4 

71.1 

94.8 

118.5 

142.2 

165.9 

189.6 

213.3 

236 

23.6 

47.2 

70.8 

94.4 

118.0 

141.6 

165.3 

188.8 

212.4 

235 

23.5 

47.0 

70.5 

94.0 

117.5 

141.0 

164.5 

188.0 

211.5 

234 

23.4 

46.8 

70.2 

93.6 

117.0 

140.4 

163.8 

187.2 

210.6 

233 

23.3 

46.6 

69.9 

93.2 

116.5 

139.8 

163.1 

186.4 

209.7 

232 

23.2 

46.4 

69.6 

92.8 

116.0 

139.2 

162.4 

185.6 

208.8 

231 

23.1 

46.2 

69.3 

92.4 

115.5 

138.6 

161.7 

184.8 

207.9 

230 

23.0 

46.0 

69.0 

92.0 

115.0 

138.0 

161.0 

184.0 

207.0 

229 

22.9 

45.8 

68.7 

91.6 

114.5 

137.4 

160.3 

183.2 

206.1 

228 

22.8 

45.6 

68.4 

91.2 

114.0 

136.8 

159.6 

182.4 

205.2 

227 

22.7 

45.4 

68.1 

90.8 

113.5 

136.2 

158.9 

181.6 

204.3 

226 

22.6 

45.2 

67.8 

90.4 

113.0 

135.6 

158.2 

180.8 

203.4 

176 


TABLE  XI.      LOGARITHMS  OF  NUMBERS. 


No.  190  L.  278.]                                  [No.  214  L.  332.  j 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Biff.  ! 

190 

278754 

8982 

9211 

9439 

9667 

9895 

1 

0123 

0351 

0578 

0806 

228 

1 

281033 

1261 

1488 

1715 

1942 

2169 

2622 

2849 

3075 

227 

2 

3301 

3527 

3753 

3979 

4205 

4431 

4656 

4882 

5107 

5332 

226 

3 

5557 

5782 

6007 

623:3 

6456 

6681 

6905 

7130 

7354 

7578 

225 

4 

7802 

8026 

8249 

847'3 

8696  jj  8920 

9143 

93G6 

9589 

9812 

223 

5 

290035 

0257 

0480 

0702 

0925 

1147 

1369 

1591 

1813 

2034 

222 

6 

2256 

2478 

2699 

2920 

3141 

3363 

3584 

3804 

4025 

4246 

221 

7 

4466 

4687 

4907 

5127 

5347 

5567 

57'87 

6007 

6226 

6446 

220 

8 

6665 

6884 

7104 

7323 

7542 

i  7761 

7979 

8198 

8416 

8635 

219 

9 

8853 

9071 

9289 

9507 

9725 

9913 

!  /Ylfti 

0378 

0595 

0813 

218 

200 

301030 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

217 

1 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

216 

2 

5351 

5566 

5781 

5996 

6211 

1  6425 

6639 

6854 

7068 

7282 

215 

3 

7496 

7710 

7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

213 

9630 

9843 

0056 

0268 

0481 

0693 

0906 

1118 

1330 

1542 

212 

5 

311754 

1966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

211 

6 

3867 

4078 

4289 

4499 

4710 

1  4920 

5130 

5340 

5551 

5760 

210 

7 

5970 

6180 

6390 

6599 

6809 

1  7018 

7227 

7436 

7646 

7854 

209 

8 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

208 

9 

320146 

0354 

0562 

0769 

0977 

i  1184 

1391 

1598 

1805 

2012 

207 

210 

2219 

2426 

2633 

2839 

3046 

3252 

3458 

3665 

3871 

4077 

206 

1 

4282 

4488 

4694 

4899 

5105 

5310 

5516 

5721 

5926 

6131 

205 

2 

6336 

6541 

67'45 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

204 

3 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

0008 

0211 

203 

4 

330414 

0617 

0819 

1022 

1225 

1427 

1630 

1832 

2034 

2236 

202 

PROPORTIONAL  PARTS. 

Biff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

225 

22.5 

45.0 

67.5 

90.0 

112.5 

135.0 

157.5 

180.0  202.5 

224 

22.4 

44.8 

67.2 

89.6 

112.0 

134.4 

156.8 

179.2 

201.6 

223 

22.3 

44.6 

66.9 

89.2 

111.5 

133.8 

156.1 

178.4 

200.7 

222 

22  2 

44.4 

66.6 

88.8 

111.0 

133.2 

155.4 

177.6  199.8 

221 

22!  1 

44.2 

66.3 

88.4 

110.5 

132.6 

154.7 

176.8 

198.9 

220 

22.0 

44.0 

66.0 

88.0 

110.0 

132.0 

154.0 

176.0 

198.0 

219 

21.9 

43.8 

65.7 

87.6 

109.5 

131.4 

153.3 

175.2 

197.1 

218 

21.8' 

43.6 

65.4 

87.2 

109.0 

130.8 

152.6 

174.4 

196.2 

217 

21.7 

43.4 

65.1 

86.8 

108.5 

ISO.  2 

151.9 

173.6 

195.3 

216 

21.6 

43.2 

64.8 

86.4 

108.0 

129.6 

151.2 

172.8 

194.4 

215 

21.5 

43.0 

64.5 

86.0 

107.5 

129.0 

150.5 

172.0 

193.5 

214 

21.4 

42.8 

64.2 

85.6 

107.0 

128.4 

149.8 

171.2 

192.6 

213 

21.3 

42.6 

63.9 

85.2 

106.5 

127.8 

149.1 

170.4 

191.7 

212 

21.2 

42.4 

63.6 

84.8 

106.0 

127.2 

148.4 

169.6 

190.8 

211 

21.1 

42.2 

63.3 

84.4 

105.5 

126.6 

147.7 

168.8 

189.9 

210 

21.0 

42.0 

63.0 

84.0 

105.0 

126.0 

147.0 

168.0 

189.0 

209 

20.9 

41.8 

62.7 

83.6 

104.5 

125.4 

146.3 

167.2 

188.1 

208 

20.8 

41.6 

62.4 

83.2 

104.0 

124.8 

145.6 

166  4 

187.2 

207 

20.7 

41.4 

62.1 

82.8 

103.5 

124.2 

144.9 

165.6 

186.3 

206 

20.6 

41.2 

61.8 

82.4 

103.0 

123.6 

144.2 

164.8 

185.4 

205 

20.5 

41.0 

C1.5 

82.0 

102.5 

123.0 

143.5 

164.0 

184.5 

204 

20.4 

40.8 

61.3 

81.6 

102.0 

122.4 

142.8 

163.2 

183.6 

203 

20.3 

40.6 

60.9 

81.2 

101.5 

121.8 

142.1 

162.4 

182.7 

202 

20.2 

40.4 

60.6 

7).  8 

101.0 

121.2 

141.4 

161.6 

181.8 

TABLE   XI.      LOGARITHMS   OF   NUMBERS. 


No.  215  L.  332.]                                  [No.  239  L.  380. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

215 

332438 

2640 

2842 

3044 

3246 

34  17 

3649 

3850 

4051 

4253 

202 

6 

4454 

4055 

4856 

5057 

5257 

5458 

5658 

5859 

6059 

6260 

201 

7 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

200 

8 

8456 

8656 

8855 

9054 

9253 

9451 

9650 

9849 

0047 

C&AC 

1QQ 

9 

340444 

0642 

0841 

1039 

1237 

1435 

1632 

1830 

2028 

2225 

iyy 

198 

220 

2423 

2620 

2817 

3014 

3212 

3409 

3606 

3802 

3999 

4196 

197 

1 

4392 

4589 

4785 

4981 

517'8 

5374 

5570 

5766 

5<J62 

6157 

196 

2 

6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

195 

3 

8305 

8500 

8694 

8889 

9083 

9278 

947'2 

9666 

9860 

0054 

1QJ. 

4 

350248 

0442 

0636 

0829 

1023 

1216 

1410 

1603 

1796 

1989 

iy-4 
193 

5 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

193 

6 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

192 

7 

6026 

6217 

6408 

6599 

6790 

6981 

7172 

7363 

7554 

7744 

191 

8 

7935 

8125 

8316 

8506 

8696 

8886 

9076 

9266 

9456 

9646 

190 

9 

9835 

0025 

0215 

0404 

0593 

0783 

nor<9 

1  1P.1 

•jq-,\ 

1  xqn 

230 

361728 

1917 

2105 

2294 

2482 

2671 

2859 

11OJ. 

3048 

3236 

looy 
3424 

189 

188 

1 

3612 

3800 

3988 

4176 

4363 

I  4551 

4739 

4926 

5113 

5301 

188 

2 

5488 

5675  |  5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

187 

3 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

186 

4 

9216 

9401 

9587 

9772 

9958 

fii  /iq 

nqoo 

A^  „ 

ftPQS 

AQOq 

-JQC 

5 

371068 

1253 

1437 

1G22 

1806 

Ui'iO 

1  1991 

Uo/to 

2175 

2360 

2544 

Uooo 

2728 

loO 

184 

6 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184 

7 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

183 

8 

6577 

6759 

6942 

7124 

7306 

7488 

767'0 

7852 

8034 

8216 

182 

9 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

38 

0030 

181 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

202 
201 

20.2 

'20.1 

40.4 
40.2 

60.6 
60.3 

80.8 
80.4 

101.0 
100.5 

121.2 
120.6 

141.4 
140.7 

161.6 
160.8 

18.1.8 
180.9 

200 

20.0 

40.0 

60.0 

80.0 

100.0 

120.0 

140.0 

160.0 

180.0 

199 

19.9 

39.8 

59.7 

79.6 

99.5 

119.4 

139.3 

159.2 

179.1 

198 

19.8 

39.6 

59.4 

79.2 

99.0 

118.8 

138.6 

158.4 

178.2 

197 

19.7 

39.4 

59.1 

78.8 

98.5 

118.2 

137.9 

157.6 

177.3 

196 

19.6 

39.2 

58.8 

78.4 

98.0 

117.6 

137.2 

156.8 

176  4 

195 

19.5 

39.0 

58.5 

78.0 

97  5 

117.0 

136.5 

156.0 

175.5 

194 

19.4 

38.8 

58.2 

77.6 

97.0 

116.4 

135.8 

155.2 

174.6 

193 

19.3 

38.6 

57.9 

77.2 

96.5 

115.8 

135.1 

154.4 

173  7 

192 

19.2 

38.4 

57.6 

'  76.8 

96.0 

115.2 

134.4 

153.6 

172.8 

191 

19.1 

38.2 

57.3 

76.4 

95.5 

114.6 

133.7 

152.8 

171.1) 

190 

19.0 

38.0 

57.0 

76.0 

95.0 

114.0 

133.  a 

152.0 

171.0 

189 

18.9 

37.8 

56.7 

75.6 

94.5 

113.4 

132.3 

151.2 

170.1 

188 

18.8 

37.6 

56.4 

75.2 

94.0 

112.8 

131.6 

150.4 

169.2 

187 

18  7 

374 

56.1 

74.8 

93.5 

112.2 

130.9 

149.6 

168.3 

186 

18.6 

37.2 

55.8 

74.4 

93.0 

111.6 

130.2 

148.8 

167.4 

185 

18.5 

37.0 

55.5 

74.0 

92.5 

111.0 

129.5 

148.0 

166.5 

184 

18.4 

36.8 

55.2 

73.6 

92.0 

110.4 

128.8 

147.2 

165.6 

183 

18.3 

36.6 

54.9 

73.2 

91.5 

109.8 

128.1 

146.4 

164.7 

182- 

18.2 

86.4 

54.6 

72.8 

1  1.0 

109.2 

127.4 

145.6 

163.8 

181 

18.1 

36.2 

54.3 

72.4 

90.5 

108.6 

126.7 

144.8 

162.9 

180 

18.0 

36.0 

54.0 

72.0 

90.0 

108.0 

126.0 

144.0 

162.0 

179 

17.9 

35.8 

53.7 

•  71.6 

89.5 

107.4 

125.3 

143.2 

161.1 

178 


TABLE  XI.     LOGARITHMS  OF  NUMBERS. 


No.  240  L.  380.] 

[No.  269  L.  431. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

S 

9 

Diff. 

240 
1 
2 
3 
4 
5 

6 

8 
9 

250 
1 

2 
3 
4 
5 

6 

7 

8 
9 

260 
1 
2 
3 

4 
5 
6 
7 
8 
9 

380211 
2017 
3815 
5606 
7390 
9166 

0392 
2197 
3995 
5785 
7568 
9343 

0573 
2377 
4174 
5964 
7746 
9520 

0754 
2557 
4.353 
6142 

7924 
9698 

0934 
2737 
4533 
6321 
8101 
9875 

1115 
2917 
4712 
6499 
8279 

1296 
3097 
4891 
6677 
8456 

1476 
3277 
5070 
6856 
8634 

1656 
3456 
5249 
7034 
8811 

1837 
3636 
5428 
7212 
8989 

181 

180 
179 
178 
178 

177 
176 
176 
175 
174 
173 

173 
172 
171 
171 
170 
169 

169 
168 
167 

167 
166 
165 

165 
164 
164 
163 
162 
162 

161 

0051 
1817 
3575 
5326 
7071 

8808 

0228 
1993 
3751 
5501 
7245 
8981 

0405 
2169 
3926 
5676 
7419 

9154 

0582 
2345 
4101 
5850 
7592 

9328 

0759 
2521 

4277 
6025 
7766 
9501 

390935 
2697 
4452 
6199 

7940 
9674 

1112 
2873 
4627 
6374 

8114 
9847 

1288 
3048 
4802 
6548 

8287 

1464 
3224 
4977 
6722 

8461 

1641 
3400 
5152 
6896 

8634 

0020 
1745 
3464 
5176 
6881 
8579 

0192 
1917 
3635 
5346 
7051 
8749 

0365 
2089 
.3807 
5517 
7221 
8918 

0538 
2261 
3978 
5688 
7^91 
9087 

0777 
2461 
4137 

5808 
7472 
9129 

0711 
2433 
4149 
5858 
7561 
9257 

0883 
2605 
4320 
6029 
7731 
9426 

1056 
2777 
4492 
6199 
7901 
9595 

1228 
2949 
4663 
6370 
8070 
9764 

401401 
3121 
4834 
6540 
8240 
9933 

411620 
3300 

4973 
6641 
8301 
9956 

1573 
3292 
5005 
6710 
8410 

0102 
1788 
3467 

5140 

6807 
8467 

0271 
1956 
3635 

5307 
6973 
8633 

0440 
2124 

3803 

5474 
7139 

8798 

0609 
2293 
3970 

5641 
7306 
8904 

0946 
2629 
4305 

5974 
7638 
9295 

1114 
2796 
4472 

6141 

7804 
9460 

1283 
2964 
4639 

6308 
7970 
9625 

1451 
3132 

4806 

6474 
8135 
9791 

0121 
1768 
3410 
5045 
6674 
8297 
9914 

0286 
1933 
3574 
5208 
6836 
8459 

0451 
2097 
3737 
5371 
6999 
8621 

0616 
2201 
3901 
5534 
7161 
8783 

0781 
2426 
4065 
5697 
7324 
8944 

0945 
2590 

4228 
5860 
7486 
9106 

1110 
2754 
4392 
6023 
7648 
9268 

1275 
2918 
4555 
6186 
7811 
9429 

1439 
3082 
4718 
6349 
7973 
9591 

421604 
3246 
4882 
6511 
8135 
9752 
43 

0075 

0236 

0398 

0559 

0720 

0881 

1042 

1203 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

178 
177 
176 
175 
174 
173 
172 
171 
170 

169 
168 
167 
166 
165 
164 
163 
162 
161 

17.8 
17.7 
17.6 
17.5 
17.4 
17.3 
17.2 
17.1 
17.0 

16.9 
16.8 
16.7 
16.6 
16.5 
16.4 
16.3 
16.2 
16.1 

35.6 
35.4 
35.2 
35.0 
34.8 
34.6 
34.4 
34.2 
34.0 

33.8 
33.6 
33.4 
33.2 
33.0 
32.8 
32.6 
32.4 
32.2 

53.4 
53.1 
52.8 
52.5 
52.2 
51.9 
51.6 
51.3 
51.0 

50.7 
50.4 
50.1 
49.8 
49.5 
49.2 
48.9 
48.5 
48.3 

71.2 
70.8 
70.4 
70.0 
69.6 
69.2 
68.8 
68.4 
68.0 

67.6 
67.2 
66.8 
66.4 
66.0 
65.6 
65.2 
64.8 
64.4 

89.0 
88.5 
88.0 
87.5 
87.0 
86.5 
86.0 
85.5 
85.0 

84.5 

84.0 
83.5 
83.0 
82.5 
82.0 
81.5 
81.0 
80.5 

106.8 
106.2 
105.6 
105.0 
104.4 
103.8 
103.2 
102.6 
102.0 

101.4 
100.8 
100.2 
99.6 
99.0 
98.4 
97.8 
97.2 
96.6 

124.6 
123.9 
123.2 
122.5 
121.8 
121.1 
120.4 
119.7 
119.0 

118.3 
117.6 
116.9 
116.2 
115.5 
114.8 
114.1 
113.4 
112.7 

142.4 

141.6 
140.8 
140.0 
139.2 
138.4 
137.6 
136.8 
136.0 

135.2 
134.4 
133.6 
132.8 
132.0 
131.2 
130.4 
129.6 
128.8 

160.2 
159.3 
158.4 
157.5 
156.6 
155.7 
154.8 
153.9 
153.0 

152.1 
151.2 
150.3 
149.4 
148.5 
147.6 
146.7 
145.8 
144.9 

TABLE   XI.      LOGARITHMS  OF   NUMBERS. 


No.  270  L  431.] 

[No.  299  L.  476. 

N. 

0 

1 

2 

3 

4 

5 

8 

7 

8 

9 

Diff. 

270 

431364 

1525 

1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

161 

1 

2969 

3130 

3290 

3450 

3610 

3770 

3930 

4090 

4249 

4409 

160 

2 

4569 

4729 

4888 

504 

8 

5207 

5367  5526 

5685 

5844 

6004 

159 

3 

6163 

6322 

6481 

6640 

6799 

6957  7116 

7275 

7433 

7592 

159 

4 

7751 

7909 

8067 

82S 

0 

8384 

8542  8701 

8859 

9017 

9175 

158 

5 

9333 

9491 

9648 

9806 

9964 

.  





, 

0122  i  n97Q 

0437 

0594 

0752 

6 

440909 

1066 

1224 

1381 

1538 

1695 

1852 

2009 

2166 

2323 

157 

7 

2480 

2637 

2793 

29£ 

0 

3106 

3263 

3419 

3576 

3732 

3889 

157 

8 

4045 

4201 

4357 

4513 

4669 

4825 

4981 

5137 

5293 

5449 

156 

9 

5604 

5760 

5915 

6071 

6226 

6382 

6537 

6692 

6848 

7003 

155 

280 

7158 

7313 

7468 

7623 

7778 

7933 

8088 

8242 

8397 

8552 

155 

8706 

8861 

9015 

91r 

'0 

9324 

9478 

9633 

9787 

9941 

0095 

154 

2 

450249 

0403 

0557 

0711 

0865 

1018 

1172 

1326 

1479 

1633 

154 

3 

1786 

1940 

2093 

224 

(7 

2400 

2553 

2706 

2859 

3012 

3165 

153 

4 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

153 

5 

4845 

4997 

5150 

53( 

)2 

5454 

5606 

5758 

5910 

6062 

6214 

152 

6 

6366 

6518 

6670 

68$ 

51 

6973 

7125 

727'6 

7428 

7579 

7731 

152 

7 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8940 

9091 

9242 

151 

g 

9392 

9543 

9694 

984 

L^ 

9995 

0146 

0296 

0447 

0597 

0748 

151 

9 

460898 

1048 

1198 

1348 

1499 

1649 

1799 

1948 

2098 

2248 

150 

290 

2398 

2548 

2697 

2847 

2997 

3146 

3296 

3445 

3594 

3744 

150 

1 

3893 

4042 

4191 

4& 

to 

4490 

4639 

4788 

4936 

5085 

5234 

149 

2 

5383 

5532 

5680 

5829 

5977   6126 

6274 

6423 

6571 

6719 

149 

3 

6868 

7016 

7164 

731 

2 

7460 

7608 

7756 

7904 

8052 

8200 

148 

4 

8347 

8495 

8643 

8790 

8938 

9085  9233 

9380 

9527 

9675 

148 

5 

9822 

9969 

0116 

02G 

,Q 

0410 

0557 

0704 

0851 

0998 

1145 

147 

6 

471292- 

1438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

146 

7 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

8 

4216 

4362 

4508 

46J 

>3 

4799 

4944 

5090 

5235 

5381 

5526 

146 

9 

5671 

5816 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

145 

L 

PROPORTIONAL  PARTS. 

Diff.   1 

2 

3 

4 

5 

6 

7 

8 

9 

161   16.1 

32.2 

48.3 

64.4 

80.5 

96.6 

112.7 

128.8 

144.9 

160   16.0 

32.0 

48.0 

64.0 

80.0 

96.0 

112.0 

128.0 

144.0 

159   15.9 

31.8 

47.7 

63.6 

79.5 

95.4 

111.3 

127.2 

143.1 

158   15.8 

31.6 

47.4 

63.2 

79.0 

94.8 

110.6 

126.4 

142.2 

157   15.7 

31.4 

47.1 

62.8 

78.5 

94.2 

109.9 

125.6 

141.3 

156   15.6 

31.2 

46.8 

62.4 

78.0 

93.6 

109.2 

124.8 

140.4 

155   15.5 

31.0 

46.5 

62.0 

77.5 

93.0 

108.5 

124.0 

139.5 

154   15.4 

30.8 

46.2 

61.6 

77.0 

92.4 

107.8 

123.2 

138.6 

153   15.3 

30.6 

45.9 

61.2 

76.5 

91.8 

107.1 

122.4 

137.7 

152   15.2 

30.4 

45.6 

60.8 

76.0 

91.2 

106.4 

121.6 

136.8 

151   15.1 

30.2 

45.3 

60.4 

75.5 

90.6 

105.7 

120.8 

135.9 

150   15.0 

30.0 

45.0 

60.0 

75.0 

90.0 

105.0 

120.0 

135.0 

149   14.9 

29.8 

44.7 

59.6 

74.5 

89.4 

104.3 

119.2 

134.1 

148   14.8 

29.6 

44.4 

59.2 

74.0 

88.8 

103.6 

118.4 

133.2 

147   14.7 

29.4 

44.1 

58.8 

73.5 

88.2 

102.9 

117.6 

132.3 

146   14.6 

29.2 

43.8 

58.4 

73.0 

87.6 

102.2 

116.8 

131.4 

145   14.5 

29.0 

43.5 

58.0 

72.5 

87.0 

101.5 

116.0 

130.5 

144   14.4 

28.8 

43.2 

57.6 

72.0 

86.4 

100.8 

115.2 

129.6 

143   14.3 

28.6 

42.9 

57.2 

71.5 

85.8 

100.1 

114.4 

128.7 

142   14.2 

28.4 

42.6 

56.8 

71.0 

85.2 

99.4 

113.6 

127.8 

141   14.1 

28.2 

42.8 

56.4 

70.5 

84.6 

98.7 

112.8 

126.9 

140   14.0 

28.0 

42.0 

56.0 

70.0 

84.0    98.0 

112.0 

126.0 

TABLE   XI.      LOGARITHMS   OF   NUMBERS. 


No.  300  L.  477.] 

[No.  339  L.  531. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

300 
1 

2 
3 

4 

5 
6 

7 
8 
9 

310 
1 
2 
3 
4 
5 
G 

8 
9 

320 
1 
2 
3 

4 
5 

6 
7 
8 
9 

330 
1 

2 
3 
4 
5 
6 
7 
8 

9 

477121 
8566 

7266 
8711 

7411 

8855 

0294 

1729 
3159 
4585 
6005 
7421 
8833 

7555 
8999 

7700 
9143 

9287 

7989 
9431 

8133 
9575 

8278 
9719 

1156 
2588 
4015 
5437 
6855 
8269 
9677 

8422 
9863 

145 
144 

144 
143 
143 

142 
142 
141 
141 

140 

140 
139 
139 
139 
138 
138 

137 
137 
186 
1S6 

136 
135 
135 

134 

134 
133 
133 
133 
132 
132 

131 

131 
131 
130 

130 
129 
129 
129 

128 
128 

480007 
1443 
2874 
4300 
5721 
7138 
8551 
9958 

0151 

1586 
3016 
4442 
5863 
7280 
8692 

0438 

1872 
3302 
4727 
6147 
7563 
8974 

0582 
2016 
3445 
4869 
6289 
7704 
9114 

0725 
2159 

3587 
5011 
6430 
7845 
9255 

0869 
2302 
3730 
5153 
6572 
7986 
9396 

1012 
2445 

3872 
5295 
6714 
8127 
9537 

1299 
2731 
4157 
5579 
6997 
8410 
9818 

0099 

1502 
2900 
4294 
5683 
7068 
8448 
9824 

0239 

1642 
3040 
4433 

5822 
7'206 
8586 
9962 

0380 

1782 
3179 
4572 
5960 
7'344 
8724 

0520 

1922 
3319 
4711 
6099 

7483 
8862 

0236 
1607 
297'3 
4335 

5693 
7046 
8395 
9740 

0661 

2062 
3458 
4850 
6238 
7621 
8999 

0374 
1744 
3109 
4471 

!  5828 
7181 
8530 
:  9874 

0801 

2201 
3597 
4989 
6376 
7759 
9137 

0941 

2341 

3737 
5128 
6515 
7897 
9275 

1081 

2481 
3876 
5267 
6653 
8035 
9412 

1222 

2621 
4015 
5406 
6791 
8173 
9550 

491362 
2760 
4155 
5544 
6930 
8311 
9687 

0099 
1470 
2837 
4199 

5557 
6911 
8260 
9606 

0511 

1880 
3246 
4607 

5964 
7316 
8664 

0648 
2017 
3382 
4743 

6099 
7451 
8799 

0785 
2154 
3518 
4878 

6234 
7586 
8934 

0922 
2291 
3655 
5014 

6370 
7721 
9068 

501059 
2427 
3791 

5150 
6505 
7856 
9203 

1196 
2564 
3927 

5286 
6640 
7991 
9337 

1333 

2700 
4063 

5421 
6776 
8126 
9471 

0009 
1349 
2684 
4016 
5344 
6668 
7987 

9303 

0143 

1482 
2818 
4149 
5476 
6800 
8119 

9434 

0745 
2053 
3356 
4656 
5951 
7243 
8531 
9815 

0277 
1616 
2951 
4282 
5609 
6932 
8251 

9566 

0411 
1750 
3084 
4415 
5741 
7064 
8382 

9697 

510545 

1883 
3218 
4548 
5874 
7196 

8514 
9828 

0679 
2017 
3351 

4681 
6006 
7328 

8646 
9959 

0813 
2151 
3484 
4813 
6139 
7460 

8777 

0947 

2284 
3617 
4946 
6271 
7592 

8909 

1081 
2418 
3750 
5079 
6403 
7724 

9040 

1215 
2551 

3883 
5211 
6535 
7855 

9171 

0090 
1400 
2705 
4006 
5304 
6598 
7888 
9174 

0221 
1530 
2835 
4136 
5434 
6727 
8016 
9302 

0353 
1661 
2966 
4266 
5563 
6856 
8145 
9430 

0712 

0484 
1792 
3096 
4396 
5693 
6985 
8274 
•  9559 

0615 
1922 
3226 
4526 
5822 
7114 
&02 
9687 

0876 
2183 
3486 
4785 
6081 
7372 
8660 
9943 

1007 
2314 
3616 
4915 
6210 
7501 
8788 

0072 
1351 

521138 
2444 
3746 
5045 
6339 
7630 
8917 

1269 
2575 
3876 
5174 
6469 
7759 
9045 

530feOO  !  0328 

0456. 

0584 

0840 

0968 

1096  !  1223 

PROPORTIONAL  PARTS. 

Piff.   1 

?      3 

4 

5 

6 

7 

8 

111.2 
110.4 
109.6 
108.8 
108.0 
107.2 
106.4 
105.6 
104.8 
104.0 
103.2 
102.4 

9 

125.1 
124.2 
123.3 
122.4 
121.5 
120.6 
119.7 
118.8 
117.9 
117.0 
116.1 
115.2 

139   13.9 
138   13.8 
137   13.7 
136   13.6 
135   13.5 
134   13.4 
133   13.3 
132   13.2 
131   13.1 
130   13.0 
129   12.9 
128  I  12.8 

27.8    41.7 
27.6    41.4 
27.4    41.1 
27.2    40.8 
27.0    40.5 
26.8    40.2 
26.6    39.9 
26.4    39.6 
26.2    89.3 
26.0    89.0 
25.8    38.7 
25.6    38.4  * 

55.6 
55.2 
54.8 
54.4 
54.0 
53.6 
53.2 
52.8 
52.4 
52.0 
51.6 
51.2 

69.5 
69.0 
68.5 
68.0 
67.5 
67.0 
66.5 
66.0 
65.5 
65.0 
64.5 
64.0 

83.4 
82.8 
82.2 
81.6 
81.0 
80.4 
79.8 
79.2 
78.6 
78.0 
77.4 
76.8 

97.3 
96.6 
95.9 
95.2 
94.5 
93.8 
93.1 
92.4 
91.7 
91.0 
90.3 
89.6 

TABLE   XI.      LOGARITHMS   OF   LUMBERS. 


No.  340  L.  531.] 

[No.  379  L.  579. 

N. 

0 

I 

2 

3 

4 

5 

e 

7 

8 

9 

Diff. 

340 
1 
2 
3 
4 
5 
6 

7 
8 
9 

350 
1 
2 

3 
4 

5 
6 

7 
8 
9 

360 
1 
2 
3 

4 
5 

6 
7 
8 
9 

370 

1 

2 
3 
4 
5 
6 

8 
9 

531479 
2754 
4026 
5294 
6558 
7819 
9076 

1607 
2882 
4153 
5421 
6685 
7945 
9202 

1734 

3009 
4280 
5547 
6811 
8071 
9327 

1862 
3136 
4407 
5674 
6937 
8197 
9452 

1990  !;  2117 
3264   3391 
4534  i  4661 
5800   5927 
7063   7189 
8322   8448 
9578  i  9703 

2245 
3518 
4787 
6053 
7315 
8574 
9829 

2372 
3645 
4914 
6180 
7441 
8699 
9954 

1205 
2452 
3696 

4936 
6172 

7405 
8635 
9861 

2500 
3772 
5041 
6306 
7567 
8825 

2627 
3899 
5167 
6432 
7693 
8951 

128 
127 
127 
126 
126 
126 

125 
125 
125 
124 

124 
124 
123 
123 

123 
122 
122 
121 
121 
121 

120 
120 
120 

119 
119 
119 
119 
118 
118 
118 

117 

117 
117 
116 
116 
116 
115 
115 
115 
114 

0079 
1330 
2576 
3820 

5060 
6296 
7529 

8758 
9984 

0204 
1454 
2701 
3944 

5183 
6419 
7652 

8881 

540329 
1579 

2825 

4068 
5307 
6543 

7775 
9003 

0455 
1704 
2950 

4192 
5431 
6666 

7898 
9126 

0580 
1829 
3074 

4316 
5555 
6789 
8021 
9249 

0705 
1953 
3199 

4440 

5678 
6913 
8144 
9371 

0830 
2078 
8388 

4564 

5802 
7036 
8267 
9494 

0955 
2203 
3447 

4688 
5925 
7159 
8389 
9616 

1080 
2327 
3571 

4812 
6049 
7282 
8512 
9739 

0106 
1328 
2547 
3762 
4973 
6182 

7387 
8589 
9787 

550228 
1450 

26G8 
3883 
5094 

6303 
7507 
8709 
9907 

0351 
1572 
2790 

4004 
5215 

6423 

7627 

8829 

0473 
1694 
2911 
4126 
5336 

6544 

7748 
8948 

0595 
1816 
3033 
4247 
5457 

6664 

7868 
9068 

0717 
1938 
3155 
43G8 
5578 

6785 
7988 
9188 

0840 
i  2060 
3276 

i  4489 
.  5699 

6905 

8108 
9308 

0962 
2181 
3398 
4610 

5820 

7026 
8228 
9428 

1084 
2303 
3519 
4731 
5940 

7146 
8349 
9548  x 

1206 
2425 
3640 
4852 
6061 

7267 
8469 
9667 

0026 
1221 
2412 
3600 
4784 
5966 
7144 

8319 
9491 

0146 
1340 
2531 
3718 
4903 
6084 
7202 

8436 

9608 

0265 
1459 
2650 
3837 
Tj021 
6202 
7319 

8554 
9725 

0385 
1578 

2709 
3955 
5139 
6320 
7497 

8671 
9o42 

!  0504 
1G98 
2887 
4074 
5257 
6437 
7014 

8788 
9^u9 

0624 
1817 
3006 
4192 
5376 
6555 
7732 

8905 

07'43 
1936 
3125 
4311 
5494 
6673 
7849 

9023 

0863 
2055 
3244 
4429 
5612 
6791 
7967 

9140 

0982 
2174 
3362 
4548 
5730 
6909 
8084 

9257 

561101 
2293 
3481 

4666 
5848 
7U2S 

8202 
9374 

0076 
1243 

2407 
3568 
4726 
5880 
7032 
8181 
9326 

0193 
1359 
2523 
3684 
4841 
5996 
7147 
8295 
9441 

0309 
1476 
2639 
.3800 
4957 
6111 
7262 
8410 
9555 

0426 
1592 
2755 
3915 
5072 
6226 
7377 
8525 
9669 

570543 
1709 
2872 
4031 
5188 
6341 
7492 
8639 

0660 
1825 
2988 
4147 
5303 
6457 
7607 
8754 

077'6 
1942 
3104 
4263 
5419 
6572 
7722 
8868 

0893 

2058 
3220 
4379 
5534 

6687 
7836 
8983 

1010 
2174 
3336 
4494 
5(550 
6802 
7951 
9097 

,  1126 
2291 
3452 
4610 

5765 
6917 
8066 
,  9212 

PROPORTIONAL  PARTS. 

Biff.   1 

2      3 

4 

5 

6 

7 

8 

9 

128   12.8 
127   12.7 
126   12.6 
125   12.5 
124   12.4 
123   12.3 
122   12.2 
121   12.1 
120   12.0 
119   11.9 

25.6    38.4 
25.4    38.1 
25.2    37.8 
25.0    37.5 
24.8    37.2 
24.6    36.9 
24.4    36.6 
24.2    36.3 
24.0    36.0 
23.8    35.7 

51.2 
50.8 
50.4 
50.0 
49.6 
49.2 
48.8 
48.4 
48.0 
47.6 

64.0 
63.5 
63.0 
62.5 
62.0 
61.5 
61.0 
60.5 
60.0 
50.5 

76.8 
76.2 
75.6 
75.0 
74.4 
73.8 
73.2 
72.6 
72.0 
71.4 

89.6 
88.9 
88.2 
87.5 
86.8 
86.1 
85.4 
84.7 
84.0 
83.3 

102.4 
101.6 
100.8 
100.0 
99.2 
98.4 
97.6 
96.8 
96.0 
95.2 

115.2 
114.3 
113.4 
112.5 
111,6 
110.7 
109.8 
108.9 
108.0 
107.1 

182 


TABLE   XI.      LOGARITHMS   OF   NUMBERS. 


No.  380.  L.  579.] 

[No.  414  L.  617. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

380 

579784 

9898 

0012 

0126 

0241 

0355  0469  Of 

>83 

0697 

0811 

114 

1 

580925 

1039 

1153 

1267 

1381 

1495 

1608  1722 

1836 

1950 

2 

2063 

2177 

2291 

2404 

2518 

2631 

2745  2858 

2972 

3085 

3 

3199 

3312 

3426 

353 

9 

3652 

3765 

3879 

31 

)92 

4105 

4218 

4 

4331 

4444 

4557 

46? 

0 

4783 

4896 

5009  5122 

5235 

5348 

113 

5 

5461 

5574 

5686 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

6 

6587 

6700 

6812 

692 

5 

7037 

7149 

7262 

7t 

574 

7486 

7599 

7 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

8608 

8720 

112 

8 

Q 

8832 
9950 

8944 

9056 

9167 

9279 

9391 

9503 

9615 

9726 

9838 

AAfil 

0173 

02&_r 

A 

0396 

0507 

0619 

0' 

-sn 

0842 

0953 

300 

591065 

UUD1 

1176 

1287 

1399 

1510 

1621 

1732 

1843 

1955 

2066 

1 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

3175 

111 

2 

3286 

3397 

3508 

361 

8 

3729 

3840 

3950 

4( 

)01 

4171 

4282 

3 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

5386 

4 

5496 

5606 

5717 

58; 

7 

5937 

6047 

6157 

6 

>67 

6377 

6487 

5 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

110 

6 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

7 

8791 

9S83 

8900 

QQQO 

9009 

9119 

9228 

9337 

9446 

9556 

9665 

9774 

ovy& 

0101 

0210 

C319 

0428 

0537 

0646 

0755 

0864 

109 

9 

600973 

1082 

1191 

1299 

1408 

1517 

1625 

1 

"34 

1843 

1951 

400 

2060 

2169 

2277 

2386 

2494 

2603 

2711 

2819 

2928 

3036 

1 

3144 

3253 

3361 

34( 

9 

3577 

!  3686 

3794 

3 

W2 

4010 

4118 

108 

2 

4226 

4334 

4442 

4550 

4658 

4766 

4874 

4 

^82 

5089 

5197 

3 

5305 

5413 

5521 

56$ 

>8 

5736 

5844 

5951 

6 

158 

6166 

6274 

4 

6381 

6489 

6596 

6704 

6811 

6919 

7026 

7133 

7241 

7348 

5 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

8205 

8312 

8419 

107 

6 

8526 
9594 

8633 
9701 

8740 
9808 

8847 
9914 

8954 

9061 

9167 

9274 

9381 

9488 

0021 

0128 

0234 

0341 

0447 

0554 

8 

610660 

0767 

0873 

0979 

1086 

1192 

1298 

1405 

1511 

1617 

9 

1723 

1829 

1936 

2043 

2148 

2254 

2360 

2466 

2572 

2678 

106 

410 

2784 

2890 

•2996 

3102 

3207 

3313 

3419 

3525 

3630 

3736 

1 

3842 

3947 

4053 

4159 

4264 

4370 

4475 

4581 

4686 

4792 

2 

4897 

5003 

5108 

521 

3 

5319 

5424 

5529 

5 

;34 

5740 

5845 

3 

5950 

6055 

6160 

6265 

6370 

6476 

6581 

6 

186 

6790 

6895 

105 

4 

7000 

7105 

7210 

7315 

7420 

7525 

7629 

7734 

7839 

7943 

PROPORTIONAL  PARTS. 

Diff.   1 

2      3 

4 

5 

6 

7 

8 

9 

118   11.8 

23.6    35.4 

47.2 

59.0 

70.8 

82.6 

94.4 

106.2 

117   11.7 

23.4    35.1 

46.8 

58.5 

70.2 

81.9 

93.6 

105.3 

116   11.6 

23.2    34.8 

46.4 

58.0 

69.6 

81.2 

92.8 

104.4 

115   11.5 

23.0    34.5 

46.0 

57.5 

69.0 

80.5 

92.0 

103.5 

114   11.4 

22.8    34.2 

45.6 

57.0 

68.4 

79.8 

91.2 

102.6 

113   11.3 

22.6    33.9 

45.2 

56.5 

67.8 

79.1 

90.4 

101.7 

112   11.2 

22.4    33.6 

44.8 

56.0 

67.2 

78.4 

89.6 

100.8 

111   11.1 

22.2    33.3 

44.4 

55.5 

66.6 

77.7 

88.8 

99.9 

110   11.0 

22.0    33.0 

44.0 

55.0 

66.0 

77.0 

88.0 

99.0 

109   10.9 

21.8    32.7 

43.6 

54.5 

65.4 

76.3 

87.2 

98.1 

108   10.8 

21.6    32.4 

43.2 

54.0 

64.8 

75.6 

86.4 

97.2 

107   10.7 

21.4    32.1 

42.8 

53.5 

64.2 

74.9 

85.6 

96.3 

106   10.6 

21.2    31.8 

42.4 

53.0 

63.6 

74.2 

84.8 

95.4 

105   10.5 

21.0    31.5 

42.0 

52.5 

63.0 

73.5 

84.0 

94.5 

105   10.5 

21.0    31.5 

42.0 

52.5 

63.0 

73.5 

84.0 

94.5 

104   10.4 

20.8    31.2 

41.6 

52.0 

62.4 

72.8 

83.2 

93.6 

TABLE  XL      LOGABTTHMS   OF  LUMBERS. 


183 


No.  415  L.  618.]                                  [No.  459  L.  662 

N. 

415 
6 

7 
8 
9 

420 
1 
2 
3 
4 
5 
6 

7 
8 
9 

430 
1 
2 
3 
4 
5 
6 

r 

8 
9 

440 
1 
2 
3 
4 
5 
6 

7 
8 
9 

450 
1 
2 
3 
4 
5 
6 
7 

8 
9 

0 

| 

2 

3 

4 

6 

6 

7 

8 

9 

Diff. 

618048 
9093 

8153 
9198 

8257 
9302 

8362 
9406 

8466 
9511 

8571 
9615 

8676 
9719 

8780 
9824 

8884 
9928 

8989 

105 
104 

103 
102 

101 
100 

99 

98 

97 
96 

95 

0032 
1072 
2110 
3146 

4179 
5210 
6238 
7263 
8287 
9308 

620136 
1176 
2214 

3249 

4282 
5312 
6340 
7366 
8389 
9410 

0240 
1280 
2318 

3353 
4^385 
5415 
6443 
7468 
8491 
9512 

0344 
1384 
2421 

3456 
4488 
5518 
6546 
7571 
8593 
9613 

0448 
1488 
2525 

3559 
4591 
5621 
6648 
7673 
8695 
9715 

0552 
1592 
2628 

3663 
4695 
5724 
6751 

7775 
8797 
9817 

0656 
1695 
2732 

3766 

4798 
5827 
6853 
7878 
8900 
9919 

0936 
1951 
2963 

3973 

4981 
5986 
0989 
7990 
8988 
9984 

0978 
1970 
2959 

3946 
4931 
5913 
6894 

7872 
8848 
9821 

0760 
1799 
2835 

3869 
4901 
5929 
6956 
7980 
9002 

0864 
1903 
2939 

3973 

5004 
6032 
7058 
8082 
9104 

0968 
2007 
3042 

4076 
5107 
6135 
7161 
8185 
9206 

0021 
1038 
2052 
3064 

4074 
5081 
6087 
7089 
8090 
9088 

0123 
1139 
2153 
3165 

4175 
5182 
6187 
7189 
JB190 
9188 

0224 
1241 
2255 
3266 

4276 

5283 
6287 
7290 
8290 
9287 

0326 
1342 
2356 
3367 

4376 

5383 
6388 
7390 
8389 
9387 

630428 
1444 
2457 

3468 
4477 
5484 
6488 
7490 
8489 
9486 

0530 
1545 
2559 

3569 
4578 
5584 
6588 
7590 
8589 
9586 

0631 
1647 
2660 

3670 
4679 
5685 
6688 
7690 
8689 
9686 

0733 

1748 
2701 

3771 

4779 
5785 
6789 
7790 
8789 
9785 

0835 
1849 
2862 

3872 
4880 
5886 
6889 
7890 
8888 
9885 

0084 
1077 
2069 
3058 

4044 
5029 
60]  1 
6992 
7969 
8945 
9919 

0183 
1177 
2168 
3156 

4143 
5127 
6110 

7089 
8007 
9043 

0283 
1276 
2267 
3255 

4242 

5226 
6*08 
7187 
8105 
9140 

0382 
1375 
2366 
3354 

4340 
5324 
6306 
7285 
8262 
9237 

640481 
1474 
2465 

3453 
4439 
5422 
6404 

7383 
8360 
9335 

0581 
1573 
2563 

3551 
4537 
5521 
6502 

7481 
8458 
9432 

0680 
1672 
2662 

3650 
4636 
5619 
6600 
7579 
8555 
9530 

0779 
1771 
2761 

3749 
4734 
5717 

6698 
7676 
8653 
9627 

0879 
1871 
2860 

3847 
4832 
5815 
6796 
7774 
8750 
9724 

0016 
0987 
1956 
2923 

3888 
4850 
5810 
6769 
7725 
8679 
9631 

0581 
1529 
2475 

0113 
1084 
2053 
3019 

3984 
4946 
5906 
6864 
7820 
8774 
9726 

0210 
1181 
2150 
3116 

4080 
5042 
6002 
6960 
7916 
8870 
9821 

650308 
1278 
2246 

3213 
4177 
5138 
6098 
7056 
8011 
8965 
9916 

660865 
1813 

0405 
1375 
2343 

3309 
4273 
5235 
6194 
7152 
8107 
9060 

0502 
1472 
2440 

3405 
4369 
5331 
6290 

7247 
8202 
9155 

0599 
1569 
2536 

3502 
4465 
5427 
6386 
7343 
8298 
9250 

0696 
1666 
2633 

3598 
4562 
5523 
6482 
7438 
8393 
9346 

0793 
1762 
2730 

3695 
4658 
5619 
6577 
7534 
8488 
9441 

0890 
1859 
2826 

3791 
4754 
5715 
6673 
7629 
8584 
9536 

0011 
0960 
1907 

0106 
1055 
2002 

0201 
1150 
2096 

0296 
1245 
2191 

0391 
1339 
2286 

0486 
1434 
2380 

0676 
1623 
2569 

0771 
1718 
2663 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

105   10.5 
104   10.4 
103   10.3 
102   10.2 
101   10.1 
100   10.0 
99    9.9 

21.0    31.5    42.0 
20.8    31.2    41.6 
20.6    30.9    41.2 
20.4    30.6    40.8 
20.2    30.3    40.4 
20.0    30.0  ,  40.0 
19.8    29.7    39.6 

52.5 
52.0 
51.5 
51.0 
50.5 
50.0 
49.5 

63.0    73.5    84.0 
62.4    72  8    83.2 
61.8    721    82.4 
61.2    71.4    81.6 
60.6    70  7    80.8 
60.0    70  0    80.0 
59.4    69.3    79.2 

94.5 
93.6 
92.7 
91.8 
90.9 
90.0 
89.1 

184 


TABLE   XI.      LOGARITHMS   OF   NUMBERS. 


No.  460  L.  662.] 

[No.  499  L.  698. 

i 
N 

0 

i 

2 

8 

4 

5 

6 

7 

8 

9 

Diff. 

460 
1 
2 
3 
4 
5 
6 
7 

8 
9 

470 
1 
2 
3 
4 
5 
6 
7 
8 

9 

480 
1 
2 
3 
4 
5 
6 
7 
8 
9 

490 
1 
2 
3 
4 
5 
6 
7 
8 
9 

662758 
3701 
4642 
5581 
6518 
7453 
8386 
9317 

2852 
3795 
4736 
5675 
6612 
7546 
8479 
9410 

2947 
3889 
4830 
5769 
6705 
7640 
8572 
9503 

3041 
3983 
4924 
5862 
6799 
7733 
8665 
9596 

3135 

4078 
5018 
5956 
6892 
7826 
8759 
9689 

3230 
4172 
5112 

6050 
6986 
7920 

8852 
9782 

3324 
4266 
5206 
6143 
7079 
8013 
8945 
9875 

3418 
4360 
5299 
6237 
7173 
8106 
9038 
9967 

3512 
4454 
5393 
6331 
7266 
8199 
9131 

3607 
4548 
5487 
6424 
7360 
8293 
9224 

94 

93 

92 
91 

90 

89 

88 
87 

0060 
0988 
1913 

2836 
3758 
4677 
5595 
6511 
7424 
8336 
9246 

0153 

1080 
2005 

2929 
3850 
4769 
5687 
6602 
7516 
8427 
9337 

670246 
1173 

2098 
3021 
3942 
4861 
5778 
6694 
7607 
8518 
9428 

0339 
1265 

2190 
3113 
4034 
4953 

5870 
6785 
7698 
8609 
9519 

0431 
1358 

2283 
3205 
4126 
5045- 
5962 
6876 
7789 
8700 
9610 

0524 
1451 

2375 
329?' 
4218 
5137 
6053 
6968 
7881 
8791 
9700 

0617 
1543 

2467 
3390 
4310 
52.28 
6145 
7059 
7972 
8882 
9791 

0710 
1636 

2560 
3482 
4402 
5320 
6236 
7151 
8063 
8973 
9882 

0802 
1728 

2652 
3574 
4494 
5412 
6328 
7242 
8154 
9064 
99T3 

0895 
1821 

2744 
3666 
4586 
5503 
6419 
7333 
8245 
9155 

0063 
0970 

1874 
2777 
3677 
4576 
5473 
6368 
7261 
8153 
9042 
9930 

0816 
1700 
2583 
3463 
4342 
5219 
6094 
6968 
7839 
8709 

0154 
1060 

1964 
2867 
3767 
4666 
5563 
6458 
7351 
8242 
9131 

0245 
1151 

2055 
2957 
3857 
4756 
5652 
6547 
7440 
8331 
9220 

680336 

1241 
2145 
3047 
3947 
4845 
5742 
6636 
7529 
8420 
9309 

0426 

1332 
2235 
3137 
4037 
4935 
5831 
6726 
7618 
8509 
9398 

0517 

1422 
2326 
3227 

4127 
5025 
5921 
6815 

7707 
8598 
9486 

0607 

1513 
2416 
3317 
4217 
5114 
6010 
6904 
7796 
8687 
9575 

0698 

1603 
2506 
3407 
4307 
5204 
6100 
6994 
7886 
8776 
9664 

0789 

1693 
2596 
3497 
4396 
5294 
6189 
7083 
7975 
8865 
9753 

0879 

1784 
2686 
3587 
4486 
5383 
6279 
717'2 
8064 
8953 
9841 

0019 

0905 
1789 
2671 
3551 
4430 
5307 
6182 
7055 
7926 
8796 

0107 

0993 
1877 
2759 
3639 
4517 
5394 
6269 
7142 
8014 
8883 

690196 
1081 
1965 
2847 
3727 
4605 
5482 
6&56 
7229 
8100 

0285 
1170 
2053 
2935 
3815 
4693 
55<59 
6444 
7317 
8188 

0373 
1258 
2142 
3023 
3903 
4731 
5657 
6531 
7404 
8275 

0462 
1347 
2230 
3111 
3991 
4868 
5744 
6618 
7491 
8362 

0550 
1435 
2318 
3199 
4078 
4956 
5832 
6706 
7578 
8449. 

0639 
1524 
2406 

3287 
4166 
5044 
5919 
6793 
7665 
8535 

0728 
1612 
2494 
3375 
4254 
5131 
6007 
6880 
7752 
8622 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6      7 

8 

9 

88.2 
87.3 
86.4 
85.5 
84.6 
83.7 
82.8 
81.9 
81.0 
80.1 
79.2 
78:3 

98 
97 
96 
95 
94 
93 
92 
91 
90 
89 
88 
87 

9.8 
9.7 
9.6 
9.5 
9.4 
9.3 
9.2 
9.1 
9.0 
8.9 
8.8 
8.7 

19.6 
19.4 
19.2 
19.0 
18.8 
18.6 
18.4 
18.2 
18.0 
17.8 
17.6 
17.4 

29.4 
29.1 

28.8 
28.5 
28.2 
27.9 
27.6 
27.3 
27.0 
26.7 
26.4 
I  26.1 

39.2 
38.8 
38.4 
38.0 
37.6 
37.2 
36.8 
36.4 
36.0 
35.6 
35.2 
34.-B  ) 

49.0 
48.5 
48.0 
47.5 
47.0 
46.5 
46.0 
45.5 
45.0 
44.5 
'44.0 
43.5 

58.8    68.6 
58.2    67.9 
57.6    67.2 
57.0    66.5 
56.4    65.8 
55.8    65.1 
55.2    64.4 
54.6    63.7 
54.0    63.0 
53.4    62.3 
52.8    61.6 
•52.2   -60.9 

78.4 
77.6 
76.8 
76.0 
75.2 
74.4 
73.6 
72.8 
72.0 
71.2 
70.4 
69.6 

TABLE   XI.      LOGARITHMS   OF   NUMBERS. 


1 85 


No.  500  L.  698.]                                 [No.  544  L.  736. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

500 

•i 

2 
3 
4 
5 
6 
7 
8 
9 

510 
1 
2 

3 
4 
5 
6 
7 
8 
9 

520 
1 
2 
3 
4 

5 
6 

7 
8 
9 

530 
1 
2 
3 
4 
5 
6 
7 

8 
9 

540 
1 
2 
3 
4 

698970 
9838 

700704 
1568 
2431 
3291 
4151 
5008 
5864 
6718 

7570 
8421 
9270 

9057 
9924 

9144 

9231 

9317 

9404 

9491 

9578 

9664 

9751 

0011 
0877 
1741 
2603 
34(53 
4322 
5179 
6035 
6888 

7710 
8591 
9440 

0098 
0963 
1827 
2689 
3549 
4408 
5265 
6120 
6974 

7826 
8676 
9524 

0184 
1050 
1913 
2775  ; 
3635 
4494  i 
5350 
62G6 
7059 

7911 
8761 
9609 

0271 
1136 
1999 
2861 
3721 
4579 
5436 
6291 
7144 

7'996 
8846 
9694 

0358 
1222 
2086 
2947 
3807 
4665 
5522 
6376 
7229 

8081 
8931 
9779 

0444 
1309 
2172 
3033 
3893 
4751 
5607 
6462 
7315 

8166 
9015 
9863 

0531 
1395 

2258 
3119 
3979 
4837 
5693 
6547 
7400 

8251 
9100 
9948 

0617 
1482 
2344 
3205 
4065 
4922 
5778 
6632 
7485 

8336 
9185 

86 

85 

84 
83 

82 
81 

80 

0790 
1654 
2517 
3377 
4236 
5094 
5949 
6803 

7655 
8506 
9355 

0033 
0879 
17'23 
2566 
3407 
4246 
5084 
5920 

6754 

7587 
8419 
9248 

710117 
0963 
1807 
2650 
3491 
4330 
5167 

6003 
6838 
7671 
8502 
9331 

0202 
1048 
1892 
2734 
3575 
4414 
5251 

6087 
6921 
7754 
8585 
9414 

0287 
1132 
1976 
2818 
3659 
4497 
5335 

6170 

7004 
7837 
8668 
9497 

0371 
1217 
2060 
2902 
3742 
4581 
5418 

6254 

7088 
7920 
8751 
9580 

0456 

1301  ; 

2144  i 
2986 
3826 
4665 
5502 

6337 
7171  j 
8003 

8834 
9663 

0540 

1385 
2229 
3070 
3910 
4749 
5586 

6421 

7254 

8086 
8917 
9745 

0625 
1470 
2313 
3154 
3994 
4833 
5669 

6504 

7338 
8169 
9000 
9828 

0710 
1554 
2397 
3238 
4078 
4916 
5753 

6588 
7421 

8253 
9083 
9911 

0794 
1639 
2481 
3323 
4162 
5000 
5836 

6671 
7504 
8336 
9165 
9994 

0077 
0903 
1728 
2552 
3374 
4194 

5013 
5830 
6646 
7460 
8273 
9084 
9893 

720159 
0986 
1811 
2634 
3456 

4276 
5095 
5912 
6727 
7541 
8354 
9165 
9974 

0242 
1068 
1893 
2716 
3538 

4358 
5176 
5993 
6809 
7623 
8435 
9246 

0055 
0863 
1669 

2474 

3278 
4079 
4880 
5679 

0325 
1151 

1975 
2798 
3620 

4440 

5258 
6075 
6890 
7704 
8516 
9327 

0136 
0944 
1750 

2555 
3358 
4160 
4960 
5759 

0407 
1233 
2058 
2881 
3702 

4522 
5340 
6156 

6972 
7785 
8597 
9408 

0217 
1024 
1830 

2635 
3438 
4240 
5040 
5838 

0490  1 
1316 
2140  i 
2963 
3784 

4604 
5422 

6238 
7053 
7866 
8678 
(J489 

0573 
1398 
2222 
3045 
3866 

4685 
5503 
6320 
7134 
7948 
8759 
9570 

0655 
1481 
2305 
3127 
3948 

4767 
5585 
6401 
7216 
8029 
8841 
9651 

0738 
1563 

2387 
3209 
4030 

4849 
5667 
6483 
7297 
8110 
8922 
9732 

0821 
1646 
2469 
3291 
4112 

49^1 

5748 
6564 
7379 
8191 
9003 
9813 

0298 
1105 
1911  ; 

2715  ; 

3518 
4320 
5120  ! 
5918 

0378 
1186 
1991 

2796 
3598 
4400 
5200 
5998 

0459 
1266 
2072 

2876 
3679 
4480 
5279 
6078 

0540 
1347 
2152 

2956 
3759 
4560 
5359 
6157 

0621 
1428 
2233 

3037 
3839 
4640 
5439 
6237 

0702 
1508 
2313 

3117 
3919 
4720 
5519 
6317 

730782 
1589 

2394 
8197 
3999 
4800 
5599 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

87    8.7 
86-   8.6 
85    8.5 
84    8.4 

17.4    26.1    34.8 
17.2    25.8    34.4 
17.0    25.5    34.0 
16.8    25.2    33.6 

43.5 
43.0 
42.5 
42.0 

52.2    60.9    69.6 
51.6    60.2    68.8 
51.0    59.5    68.0 
50.4    58.8    67.2 

78.3 
77.4 
76.5 
75.  6j 

TABLE   XI.      LOGARITHMS   OF   NUMBERS. 


No.  545  L.  736.] 

[No.  584  L.  707. 

N. 

0 

1 

2 

3 

4     5 

6 

7 

8 

9   Diff. 

545~ 

736397  6476 

6556 

6635  6715   6795 

6874 

6954 

7034  7113 

6 

7193  7272 

7352 

743 

1 

7511   7590 

7670 

7 

749 

7829  7908 

7 

7987 

8067 

8146 

8225 

8305 

8384 

8463 

8543 

8622  8701 

8 

8781 

8860  8939 

9018 

9097 

1  9177 

9256 

9335 

9414 

9493 

9 

9572 

9651  '  9731 

981 

Q 

9889 

9968 

0047 

0126 

0205 

0284 

79 

550 

740363 

0442 

0521 

0600 

0678 

0757 

0836 

0915 

0994 

1073 

1 

1152 

1230 

1309 

138 

8 

1467 

1546 

1624 

1 

703 

1782 

1860 

2 

1939 

2018 

2096 

21? 

5 

2254 

2332 

2411 

2489 

2568 

2647 

3 

2725 

2804 

2882 

2961 

3039 

3118 

3196 

3275 

3353 

3431 

4 

3510 

3588 

3667 

374 

5 

3823 

3902 

3980 

4 

058 

4136 

4215 

5 

4293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

6 

507'5 

5153 

5231 

5309 

5387 

i  5465 

5543 

5621 

5699 

5777 

78 

7 

5855 

5933 

6011 

60S 

9 

6167 

1  6245 

6323 

6* 

401 

6479 

6556 

8 

6634 

6712 

6790 

68C 

8 

6945 

7023  |  7101 

7179 

7256 

7334 

9 

7412 

7489 

7567 

7645 

7722 

7800  !  7878 

7955 

8033 

8110 

560 

8188 

8266 

8343 

8421 

8498 

8576  8653 

8731 

8808 

8885 

1 

8963 

9040 

9118 

91S 

5 

9272 

i  9350  9427 

9 

504 

9582 

9659 

g 

9736 

9814 

9891 

996 

8 

0045 

0123  0200 

0 

077 

0354 

0431 

3 

750508 

0586 

0663 

0740 

0817 

0894  0971 

1048 

1125 

1202 

4 

1279 

1356 

1433 

151 

0 

1587 

1604  i  1741 

1 

818 

1895 

1972 

5 

2048 

2125 

2202 

2279 

2356 

2433  2509 

2586 

2663 

2740 

77 

6 

2816 

2893 

2970 

3047 

3123 

3200  3277 

3353 

3430 

3506 

fi 

3583 

3660 

3736 

381 

8 

3889 

3966  4042 

4 

119 

4195 

4272 

8 

4348 

4425 

4501 

4578 

4654 

4730  4807 

4883 

4960 

5036 

9 

5112 

5189 

5265 

5341 

5417 

5494 

5570 

5646 

5722 

5799 

570 

5875 

5951 

6027 

6103 

6180 

6256 

6332 

6408 

6484 

6560 

1 

6636 

6712 

6788 

686 

>4 

6940 

7016 

7093 

7 

168 

7244 

7320 

76 

2 

7396 

7472 

7548 

76$ 

J4 

7700 

7775 

7851 

7 

927 

8003 

8079 

3 

8155 

8230 

8306 

8& 

)2 

8458 

8533 

8609 

8685 

8761 

8836 

4 

8912 

8988 

9063 

911 

59 

9214 

9290 

9366 

9 

441 

9517 

9592 

5 

9668 

9743 

9819 

98! 

4 

9970 

i  0045 

0121 

0196 

0272 

0347 

-  6 

760422 

0498 

0573 

0649 

0724 

0799 

0875 

0950 

1025 

1101 

7 

1176 

1251 

1326 

14( 

)2 

1477 

1552 

1627 

1 

702 

1778 

1853 

8 

1928 

2003 

2078 

2153 

2228 

!  2303 

2378 

2453 

2529 

2604 

9 

26/9 

2754 

2829 

2904 

2978 

3053 

3128 

3203 

3278 

3353 

75 

580 

3428 

3503 

3578 

36£ 

>3 

3727 

3802 

3877 

3952 

4027 

4101 

1 

4176 

4251 

4326 

4400 

4475 

4550 

4624 

4699 

4774 

4848 

2 

4923 

4998 

5072 

51^ 

(7 

5221 

5296 

5370 

5 

445 

5520 

5594 

3 

5669 

5743 

5818 

5892 

5966 

6041 

6115 

6190 

6264 

6338 

4 

6413 

6487 

6562 

6& 

56 

6710 

6785 

6859 

6933 

7007 

7082 

PROPORTIONAL,  PARTS. 

Diff.   1 

2      8 

4 

5 

6 

7     8 

9 

83    8.3 

16.6    24 

.9 

33.2 

41.5 

49.8 

58.1    66.4 

74.7 

82    8.2 

16.4    24.6 

32.8 

41.0 

49.2 

57.4    65.6 

73.8 

81    8.1 

16.2    24.3 

32.4 

40.5 

48.6 

56.7    64.8 

72.9 

80    8.0 

16.0    24.0 

32.0 

40.0 

48.0 

56.0    64.0 

72.0 

79    7.9 

15.8    23.7 

31.6 

39.5 

47.4 

55.3    63.2 

71.1 

78    7.8 

15.6    23.4 

31.2 

39.0 

46.8 

54.6    62.4 

70.2 

77    7.7 

15.4    23.1 

30.8 

38.5 

46.2 

53.9    61.6 

69.3 

76    7.6 

15.2    22.8 

30.4 

38.0 

45.6 

53.2    60.8 

68.4 

75    7.5 

15.0    22.5 

30.0 

37.5 

45.0 

52.5    60.0 

67.5 

74    7.4 

14.8    22.2 

29.6 

37.0 

44.4 

51.8    59.2 

66.6 

TABLE   XI.      LOGARITHMS   OF  NUMBERS. 


187 


No.  585  L.  767.]                                  [No.  629  L.  799. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

5S5 

767156 

7230 

7304 

7379 

7453 

7527  7601 

7675 

7749 

7823 

6 

7898 

7972 

8046 

8120 

8194 

!  8268  8342 

8416 

8490 

8564 

74 

7 

8638 

8712 

8786 

8860 

8934 

9008  9082 

9156 

9230  |  9303 

8 

9377 

9451 

9525 

9599 

9673 

9746  9820 

9894 

9968  !  

!  nruo 

9 

770115 

0189 

0263 

0336 

0410 

0484 

0557 

0631 

0705 

0778 

590 

0852 

0926 

0999 

1073 

1146 

1220 

1293 

1367 

1440 

1514 

1 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

2 

2322 

2395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

3 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

4 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

5 

4517 

4590 

4663 

4736 

4809 

!  4882 

4955 

5028 

5100 

5173 

6 

5246 

5319 

5392 

5465 

5538 

5610 

5683 

5756 

5829 

5902 

7 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

8 

6701 

6774 

6S46 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

9 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

600 

8151 

8224 

8296 

8368 

8441 

!  8513 

8585 

8658 

8730 

8802 

1 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

2 

9596 

9669 

9741 

9813 

9885 

9957 

0029 

0101 

0173 

AOJS 

3 

780317 

0389 

0461 

0533 

0605 

0677 

0749 

0821 

0893 

UiC'iO 

0965 

72 

4 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

5 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

6 

2473 

2544 

2616 

2688 

2759 

2831 

2902 

297'4 

3046 

3117 

7 

3189 

3260 

3332 

3403 

3475 

I  3546 

3618 

3689 

3761 

3832 

8 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

9 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

610 

5330 

5401 

5472 

5543 

5615 

5686 

5757 

5828 

5899 

5970 

1 

6041 

6112 

6183 

6254 

6325 

|  6396 

6467 

6538 

6609 

6680 

71 

2 

6751 

6822 

6893 

6964 

7035 

;  71  u6 

7177 

7248 

7319 

7390 

3 

7460 

7531 

7602 

7673 

7744 

78i5 

7885 

7956 

8027 

8098 

4 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

5 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

g 

9581 

9651 

9722 

9792 

9863 

9933 

0004 

0074 

0144 

0215 

7 

790285 

0356 

0426 

0496 

0567 

0637 

0707 

0778 

0848 

0918 

8 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1550 

1620 

9 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

620 

2392 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

70 

1 

3092 

3162 

3231 

3301 

3371 

3441 

3511 

3581 

3651 

3721 

2 

3790 

3860 

3930 

4000 

4070 

4139 

4209 

4279 

4349 

4418 

3 

4488 

4558 

4627 

4697 

4767 

i  4836 

4906 

4976 

5045 

5115 

4 

5185 

5254 

5324 

5393 

5463 

5532  5602 

5672 

5741 

5811 

5 

5880 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

6 

6574 

6644 

6713 

6t82 

6852 

6921 

6990 

7060 

7129 

7198 

7 

7268 

7337 

7406 

7475 

7545 

1  7614 

7683 

7752 

7821 

7890 

8 

7960 

8029 

8098 

8167 

8236 

8305  8374 

8443 

8513 

8582 

9 

8651 

8720 

8789 

8858 

8927 

!  8996 

9065 

9134 

9203 

9272 

69 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

6     7     8 

9 

75    7.5 

15.0    22.5    30.0 

37.5 

45.0    52.5    60.0 

67.5 

74    7.4 

14.8    22.2    29.6 

37.0 

44.4    51.8    59.2 

66.6 

73    7,3 

14.6    21.9    29.2 

36.5 

43.8    51.1    58.4 

65.7 

72    7.2 

14.4    21.6    28.8 

36.0 

43.2    50.4    57.6 

64.8 

71    7.1 

14.2    21.3    28.4 

35.5 

42.6    49.7    56.8 

63.9 

70    7.0 

14.0    21.0    28.0 

35.0 

42.0    49.0    56.0 

63.0 

;   69    6.9 

13.8    20.7    27.6 

34.5 

41.4    48.3    55.2 

62.1 

> 

188 


TABLE   XI.      LOGARITHMS   OF    NUMBERS. 


—  — 
No.  630  L.  799.]                                [No.  674  L.  829. 

N. 

0 

1 

2 

3 

Ll  5 

6 

7 

8 

9 

Diff. 

630 

799341 

9409 

9478 

9547 

9616   9685 

9754 

9823 

9892 

9961 

1 

800029 

0098 

0167 

0236 

0305   0373 

0442 

0511 

0580 

0648 

2 

0717 

0786 

0854 

0923 

0992  !  1061 

1129 

1198 

1266 

1335 

3 

1404 

1472 

1541 

1609 

1678  I!  1747 

1815 

1884 

1952 

2021 

4 

2089 

2158 

2226 

2295 

2363   2432 

2500 

2568 

2637 

2705 

5 

2774 

2842 

2910 

2979 

3047   3116 

3184 

3252 

3321 

3389 

6 

3457 

3525 

3594 

3662 

3730  1  3798 

3867 

3935 

4003 

4071 

7 

4139 

4208 

4276 

4344 

4412  i 

4480 

4548 

4616 

4685 

4753 

8 

4821 

4889 

4957 

5025 

5093 

51G1 

5229 

5297 

5365 

5433 

68 

9 

5501 

5569 

5637 

5705 

5773 

5841 

5908 

5976 

6044 

6112 

640 

806180 

6248 

6316 

6384 

6451 

6519 

6587 

6655  6723 

6790 

1 

6858 

6926 

6994 

7061 

7129 

7197 

7264 

7332  7400 

7467 

2 

7535 

7603 

7670 

7738 

7'806 

7873 

7941 

8008 

8076 

8143 

3 

8211 

827'9 

8346 

8414  1  8481 

8549 

8616 

8684 

8751 

8818 

4 

8886 

8953 

9021 

9088 

9156 

9223 

9290 

9358 

9425 

9492 

5 

9560 

9627 

9694 

9762 

9829 

9896 

9964 

0031 

0098 

0165 

6 

810233 

0300 

0367 

0434 

0501 

0569 

0636 

0703 

0770 

0837 

7 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

67 

8 

1575 

1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

9 

2245 

2312 

2379 

2445 

2512 

2579 

2646 

2713 

2780 

2847 

650 

2913 

2980 

3047 

3114 

3181 

3247 

3314 

3381 

3448 

3514 

1 

3581  , 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

2 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

3 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

5378 

5445 

5511 

4 

5578 

5644 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

5 

6241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6&S8 

6 

6904 

6970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

r< 

7565 

7631 

7698 

7764 

7830 

7896 

7'962 

8028 

8094 

8160 

8 

8226 

8292 

8358 

8424 

8490 

"8556 

8622 

8688 

8754 

8820 

fifi 

9 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

DO 

660 

9544 

9610 

9676 

9741 

9807 

9873 

9939 

L 

0004 

0070 

0136 

1 

820201 

0267 

0333 

0399 

0464 

0530 

0595 

0661 

0727 

0792 

2 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

3 

1514 

1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

4 

2168 

2233 

2299 

23(M 

2430 

2495 

2560 

2626 

2691 

2756 

5 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

6 

3474 

3539 

3605 

3670 

3735 

3800 

3865 

3930 

3996 

4061 

4126 

4191 

4256 

4321 

4386 

4451 

4516 

4581 

4646 

4711 

65 

8 

477(5 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

5296 

5361 

9 

5426 

5491 

5556 

5621 

5686 

5751 

5815 

5880 

5945 

6010 

670 

6075 

6140 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

1 

6723 

6787 

6852 

6917 

6981 

7046 

7111 

7175 

7240 

7805 

2 

7369 

74:34 

7499 

7563  7628 

7692 

7757 

7821 

7886 

7951 

3 

8015 

8080 

8144 

8209 

8273 

8338 

8402 

8467 

8531  8.yJ5 

4 

8660 

8724 

8789 

8853 

8918   8982 

9046 

9111 

9175  i  9239 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

68    6.8 

13.6    20.4  !  27.2 

34.0 

40.8    47.6    54.4 

61.2 

67    6.7 

13.4  I  20.1  i  26.8 

33.5 

40.2    46.9    53.6 

60.3 

66    6.6 

13.2    19.8    26.4 

33.0 

39.6    46.2    52.8 

59  4 

65    6.5 

13.0    19.5    26.0 

32.5 

39.0    45.5    52.0 

58.5 

64    6.4 

1£.8    19.2    25.6 

32.0 

38.4    44.8    51.2 

57.6 

TABLE  XI.      LOGARITHMS  OF   NUMBERS. 


189 


No.  675  L.  829.]                                 [No.  719  L.  857. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

675 

829304 

9368 

9432 

9497 

9561 

9625 

9690 

9754 

9818 

9882 

Q 

9947 

0011 

0075 

0139  '''  nortl 

;  0268 

0332 

OSQfi 

0460 

0525 

7 

830589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

8 

1230 

1294 

1358 

1422 

1486 

1550 

1614 

1678 

1742 

1806 

64 

9 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

680 

2509 

2573 

2637 

2700 

2764 

2828 

2892 

2956 

3020 

3083 

1 

3147 

3211 

3275 

3338 

3402 

3466 

3530 

3593 

3657 

3721 

2 

3784 

3848 

3912 

3975 

4039 

i  4103 

4166 

4230 

4294 

4357 

3 

4421 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

4 

5056 

5120 

5183 

5247 

5310 

!  5373 

5437 

5500 

5564 

5627 

5 

5691 

5754 

5817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

6 

6324 

6387 

6451 

6514 

6577 

i  6641 

6704 

6767 

6830 

6894 

7 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

8 

7588 

7652 

7715 

7778 

7841 

•  7904 

7967 

8030 

8093 

8156 

9 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

63 

490 

8849 

8912 

8975 

9038 

9101 

9164 

9227 

9289 

9352 

9415 

9478 

9541 

9604 

9667 

Q79Q 

9792 

9855 

9918 

9981 

0043 

2 

840106 

0169 

0232 

0294 

0357 

|  0120 

0482 

0545 

0608 

0671 

3 

0733 

0796 

0859 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

4 

1359 

1422 

1485 

1547 

1610 

1672 

1735 

1797 

1860 

1922 

5 

1985 

2047 

2110 

2172 

2235 

2297 

2360 

2422 

2484 

2547 

6 

2609 

2672 

2734 

2796 

2859 

2921 

2983 

3046 

3108 

3170 

7 

3233 

3295 

3357 

3420 

3482 

3544 

3606 

3669 

3731 

3793 

8 

3855 

3918 

3980 

4042 

4104 

4166 

4229 

4291 

4353 

4415 

9 

4477 

4539 

4601 

4664 

4726 

4788 

4850 

4912 

4974 

5036 

700 

5098 

5160 

5222 

5284 

5346 

5408 

5470 

5532 

5594 

5656 

62 

1 

5718 

5780 

5842 

5904 

5966 

6028 

6090 

6151 

6213 

6275 

2 

6337 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6&S2 

6894 

3 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

4 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004  !  8066  !  8128 

5 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

8620  8682  ;  8743 

6 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235  9297 

9358 

7 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849  9911 

9972 

8 

850033 

0095 

OJ56 

0217 

0279 

0340 

0401 

0462  0524 

0585 

9 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075  1136 

1197 

710 

1258 

1320 

1381 

1442 

1503 

1564 

1625 

1686  1747 

1809 

1 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297  1  2358  2419 

2 

2480 

2541 

2«02  2663 

2724 

2785 

2846 

2907  !  2968  !  3029 

61 

8 

8090 

3150 

3211  327'2 

3333 

3394  [  3455 

3516  3577  3637 

4 

3698 

S759 

3820  3881 

3941 

4002  |  4063 

4124  !  4185  ;  4245 

5 

4306 

4367 

44-28  4488 

4549 

4610 

4670 

4731  :  4792  4852 

6 

4913 

4974 

5034  5095 

5156 

5216 

5277 

5337  5398  5459 

7 

5519 

5580 

5640  5701 

5761 

5822 

5882 

5943  6003  6064 

8 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548  6608  6668 

9 

6729 

6789 

6850 

6910 

6970  ; 

7031 

7091 

7152 

7212 

7272 

\ 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

65    6.5 

13.0    19.5    26.0 

32.5 

39.0    45.5    52.0 

58.5 

64    6.4 

12.8    19.2    25.6 

32.0 

38.4    44.8    51.2 

57.6 

63    6.3 

12.6    18  9    25.2 

31.5 

37.8    44.1    50.4 

56.7 

62    6.2 

12.4    18.6    24.8 

31.0 

37.2    43.4    49.6 

55.8 

61    6.1 

12.2    18.3    24.4 

30.5 

36.6    42.7    48.8 

54  9 

60    6.0 

12.0    18.0    24.0 

30.0 

36.0    42.0    48.0 

54.0 

190 


TABLE   XI.      LOGARTTHMS   OF   NUMBERS. 


No.  720  L.  867.]                                 [No.  764  L.  883. 

. 

Diff. 

720 

857332 

7393 

7453 

7513 

7574 

7634 

7694 

7755 

7815 

7875 

1 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

2 

8537 

8597  8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

3 

9138 

9198  9258 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

60 

4 

9739 

9799 

9859 

9918 

9978 

0038 

0098 

0158 

0218 

0278 

5 

860338 

0398 

0458 

0518 

0578 

0637 

0697 

0757 

0817 

0877 

6 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

7 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

8 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

9 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3114 

3204 

3263 

730 

3323 

3382 

3442 

3501 

3561 

3620 

3680 

3739 

3799 

3858 

1 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

2 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

3 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

5519 

5578 

5637 

4 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

5 

6287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

6 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

50 

7 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

8 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8586 

9 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

91  ,  3 

740 

9232 

9290 

9349 

9408 

9466 

9525 

9584 

9642 

9701 

9760 

1 

9818 

9877 

9935 

9994 

0053 

0111 

0170 

0228 

09S7 

0345 

2 

870404 

0462 

0521 

0579 

06:38 

0696 

0755 

0813 

UCO4 

0872 

0930 

3 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

4 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2008 

5 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

6 

2739 

2797 

2855 

2913 

2972 

3030 

3088 

3146 

3204 

3262 

7 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

8 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

58 

9 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4945 

5003 

750 

5061 

5119 

5177 

5235 

5293 

5351 

5409 

5466 

5524 

5582 

1 

5640 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

2 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

3 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

4 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

5 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

6 

8522 

8579 

8637 

8694 

8752 

i  8809 

8866 

8924 

8981 

9039 

7 

9096 

9153 

9211 

9268 

9325 

|  9383 

9440 

9497 

9555 

9612 

g 

9669 

9726 

9784 

9841 

9898 

9956 

0013 

0070 

0127 

0185 

9 

880242 

0299 

0356 

0413 

0471 

0528 

0585 

0642 

0699 

0756 

760 

0814 

0871 

0928 

0985 

1042 

1099 

1156 

1213 

1271 

1328 

1 

1385 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

2 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

57 

3 

2525 

2581 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3037 

4 

3093 

3150 

32C7 

3264 

3321 

3377 

3434 

3491 

3548 

3605 

PROPORTIONAL  PARTS. 

Diff 

1 

2 

3      4 

5 

678 

9 

59 

5.9 

11.8 

17.7    23.6 

29.5 

35.4    41.3    47.2 

53.1 

58 

5.8 

11.6 

17.4    23.2 

29.0 

34.8    40.6    46.4 

52.2 

57 

5.7 

11.4 

17.1    22.8 

28.5 

34.2    39.9    45.6 

51.3 

56 

5.6 

11.2 

16.8    22.4 

28.0 

33.6    89.2    44.8 

50.4 

TABLE    XI.      LOGARITHMS   OF    NUMBERS. 


191 


No.  765  L.  883.]                                 [No.  809  L.  908. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

765 

883661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4115 

4172 

6 

4229 

4285 

4342 

4399 

4455 

4512 

4569 

4625 

4682 

4739 

7 

4795 

4852 

4909 

4965 

5022 

5078 

5135 

5192 

5248 

5305 

8 

5361 

5418 

5474 

5531 

5587 

5644 

5700 

5757 

5813 

5870 

9 

5926 

5983 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

770 

6491 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942 

6998 

1 

7054 

7111 

7167 

7223 

7280 

7336 

7392 

7449 

7505 

7561 

2 

7617 

7674 

7730 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

3 

8179 

8236 

8292 

SMS 

8404 

8460 

8516 

8573 

8629 

8685 

4 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

5 

g 

9302 

9862 

9358 
9918 

9414 

9974 

9470 

9526 

9582 

9638 

9694 

9750 

9806 

56 

0030 

0086 

0141 

0197 

0253 

0309 

0365 

7 

890421 

G47'7 

0533 

0589 

0645 

0700 

0756 

0812 

0868 

0924 

8 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

9 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

780 

2095 

2150 

2206 

2262 

2317 

2373 

2429 

2484 

2540 

2595 

1 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

2 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

3 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

4 

4316 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

4814 

5 

4870 

4925 

4980 

5036 

5091 

5146 

5201 

5257 

5312 

5367 

6 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

5975 

6030 

6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

8 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

9 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

55 

790 

7627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8067 

8122 

1 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

2 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

3 

9273 

QR91 

9328 

nor'K 

9383 

QQOA 

9437 

QQQC 

9492 

9547 

9602 

9656 

9711 

9766 

4 

Jo<4L 

yof  o 

vJo\J 

yyoo 

0039 

0094 

0149 

0203 

0258 

0312 

5 

900367 

0422 

0476 

0531 

0586 

0640 

0695 

0749 

0804 

0859 

6 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

7 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

8 

2003 

2057 

2112 

21  6£ 

2221 

2275 

2329 

2384 

2438 

2492 

9 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

800 

3090 

3144 

3199 

3253 

3307 

3361 

3416 

3470 

3524 

3578 

1 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

2 

4174 

4229 

4283 

4387 

4391 

4445 

4499 

4553 

4607 

4661 

3 

4716 

4770 

4824 

4873 

4932 

4986 

5040 

5094 

5148 

5202 

54 

4 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

5 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

6 

6335 

6389 

0443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

8 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

9 

7949 

8002 

8006 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

PROPORTIONAL  PARTS. 

Difif.   1 

234 

5 

678 

9 

57    5.7 

11.4    17.1    22.8 

28.5 

34.2    39.9    45.6 

51.3 

56    5.6 

11.2    16.8    22.4 

28.0 

33.6    39.2    44.8 

50.4 

55    5.5 

11.0    16.5    22.0 

27.5 

33.0    38.5    44.0 

49.5 

54    5.4 

10.8    16.2    21.6 

27.0 

32.4    37.8    43.2 

48.6 

TABLE   XI.      LOGARITHMS   OF   NUMBERS. 


No.  810  L.  908.]                                 LNo.  854  L.  93l. 

N. 

0 

1 

2 

3 

4    5 

6 

7 

8 

9 

Dift 

i 

810 

908485 

8539 

8592 

8646 

8699   8753 

8807  8860 

8914 

8967 

1 

9021 

9074 

9128 

9181 

9235  !  9289 

9342 

9396 

9449  9503 

2 

9556 

9610 

9663 

9716 

9770   9823 

9877 

9930 

ooej.  i 
rwio'y 

3 

910091 

0144 

0197 

0251 

0304  i  0358 

0411   0464 

0518 

WUI 

0571 

4 

0624 

0678 

0731 

0784 

0838   OH91 

0944 

0998 

1051 

1104 

5 

1158 

1211 

1264 

1317 

1371   1424 

1477 

1530 

1584 

1637 

6 

1690 

1743 

1797 

1850 

1903   1956 

2009 

2063 

2116 

2169 

7 

2222 

2275 

2328 

2381 

2435  i  2488 

2541 

2594 

2647 

2700 

8 

2753 

2806 

2859 

2913 

2966  i  3019 

3072 

3125 

3178 

3231 

9 

3284 

3337 

3390 

3443 

3496   3549 

3602 

3655 

3708 

3761 

63 

830 

3814 

3867 

3920 

3973 

4026  i  4079 

4132 

4184 

4237 

4290 

1 

4343 

4396 

4449 

4502 

4555  ;  4608 

4660 

4713 

4766 

4819 

2 

4872 

4925 

4977 

5030 

5083   5136 

5189 

5241   5294 

5347 

3 

5400 

5453 

5505 

5558 

5611   5664 

5716 

5769  j  5822 

5875 

4 

5927 

5980 

6033 

6085 

6138   6191 

6243 

6296  6349 

6401 

5 

6454 

6507 

6559 

6612 

6664   6717 

6770 

(822 

6875 

6927 

6 

6980 

7033 

7085 

7138 

7190   7243 

7295 

7348 

7400 

7453 

7 

7506 

7558 

7611 

7663 

7716   7768 

7820 

7873 

7925 

7978 

8 

8030 

8083 

8135 

8188 

8240  i  8293 

8345 

8397 

8450 

8502 

9 

8555 

8607 

8659 

8712 

8764   8816 

8869 

8921 

8973 

9026 

830 
j 

9078 
9601 

9130 
9653 

9183 
9706 

9235 

9758 

9287  '  9340 

QO-l  0  i   OQftO 

9392 

QQ14 

9444 

9967 

9496 

9549 

yoiu 

S7UU,* 

yyii 

1  nn-io  1  nr^-i 

2 

920123 

0176 

0228 

0280 

0332 

0384 

0436 

0489 

0541 

0593 

3 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

£A 

4 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

OH 

5 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

6 

2200 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

7 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

8 

3244 

3296 

3348 

3399 

3451 

3503 

3555 

3607 

3658 

3710 

9 

3762 

3814 

3866 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

840 

4279 

4331 

4383 

4434 

4486 

4538 

4589 

4641 

4693  !  4744 

1 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

5261 

2 

5312 

5364 

5415 

5467 

5518 

5570 

5621 

5673 

5725 

5776 

3 

5828 

5879 

5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

4 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

5 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

6 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

7 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

8 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

9 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9368 

850 

9419 

9470 

9521 

9572 

9623 

9674 

9725 

9776 

9827 

9879 

j 

9930 

9981 

51 

0032 

0083 

0134 

0185 

0236 

0287 

0338 

0389 

2 

930440 

0491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

3 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

4 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

53    5.3 

10.6    15.9    21.2 

26.5 

31.8    37.1    42.4 

47.7 

52    5.2 

10.4    15.6    20.8 

26.0 

31.2    36.4    41.6 

46.8 

51    5.1 

10.2    15.3    20.4 

25.5 

30.6    35.7    40.8 

45.9 

50    5.0 

10.0    15.0    20.0 

25.0 

30.0    35.0    40.0 

45.0 

TABLE   XI.      LOGARITHMS   OF   NUMBERS. 


193 


No.  855  L.  931.]                                  [No.  899  L.  954. 

K, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

355 

931966 

2017 

2068 

2118 

2169 

2220  2271 

2322 

2372 

2423 

6 

2474 

2524 

2575 

2626 

2677 

2727  i  2778 

2829 

2879 

2930 

7 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

8 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

9 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

860 

4498 

4549 

4599 

4650 

4700 

4751 

4801 

4852 

4902 

4953 

1 

5003 

5054 

5104 

5154 

5205 

5255 

5306 

5356 

5406 

5457 

2 

5507 

5558 

5608 

5658 

5709 

5759 

5809 

5860 

5910 

5960 

3 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

4 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

5 

7016 

7066 

7116 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

6 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

7 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

50 

8 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

9 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

870 

9519 

9569 

9619 

9669 

9719 

9769 

9819 

9869 

9918 

9968 

1 

940018 

0068 

0118 

0168 

0218 

0267 

0317 

0367 

0417 

0467 

2 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0915 

0964 

3 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

4 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909 

1958 

5 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

6 

2504 

2t>54 

2603 

2653 

2702 

2752 

2801 

2851 

2901 

2950 

7 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

8 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

-3890 

3939 

9 

3989 

4038 

4088 

4137 

4186 

4236 

4286 

4335 

4384 

4433 

880 

4483 

4532 

4581 

4631 

4680 

4729 

4779 

4828 

4877 

4927 

1 

4976 

5025 

5074 

5124 

5173 

5222 

5272 

5321 

5370 

5419 

2 

5469 

5518 

5567 

5616 

5665 

5715 

5764 

5813 

5862 

5912 

3 

5961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

4 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

5 

6943 

6992 

7041 

7090 

7139 

7189 

7238 

7287' 

7336 

7385 

6 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7875 

49 

7 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

8 

8413 

8462 

8511 

8560 

8608 

8657 

8706 

8755 

8804 

8853 

9 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

390 
1 

9390 

9878 

9439 
9926 

9488 
9975 

9536 

9585 

9634 

9683 

9731 

9780 

9829 

0024 

0073 

0121 

0170 

0219 

0267 

0316 

2 

950365 

0414 

0462 

0511 

0560 

0608 

0657 

0706 

0754 

0803 

3 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

4 

1338 

1386 

1435 

1483 

1532 

1580 

1629 

1677 

1726 

1775 

5 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

6 

2308 

2356 

2405 

2453 

2502 

2550 

2599 

2647 

2696 

2744 

7 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

8 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

9 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

PROPORTIONAL  PARTS. 

Diff 

1 

2 

3      4 

5 

678 

9 

51 

5.1 

10.2 

15.3    20.4 

25.5 

30.6    35.7    40  8 

45  9 

50 

5.0 

10.0 

15.0    20.0 

25.0 

30.0    35.0    40.0 

45.0 

49 

4.9 

9.8 

14.7    19.6 

24.5 

29.4    34.3    39.2 

44  1 

48 

4.8 

9.6 

14.4    19.2 

24.0 

28.8    33.6    38.4 

43.2 

194 


TABLE   XI.      LOGARITHMS   OF   NUMBERS. 


No  900  L.  954.1                                 [No.  944  L.  975. 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Diff. 

900 

954243 

4291 

4339 

4387 

4435 

4484 

4532 

4580 

4628 

4677 

1 

4725 

4773 

4821 

4869 

4918 

4966 

5014 

5062 

5110 

5158 

2 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

3 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

4 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6553 

6601 

5 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

48 

6 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

7559 

7 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

8 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

9 

8564 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

910 

9041 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

9471 

1 

9518 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

2 

9995 

0042 

0090 

0138 

0185 

0233 

0280 

0328 

0876 

0423 

3 

960471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

4 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

5 

1421 

1469 

1516 

1563 

1611 

1058 

17'06 

1753 

1801 

1848 

6 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

7 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

8 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

9 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

920 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

1 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

2 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061   5108 

5155 

3 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

5578 

5625 

4 

5672 

5719 

5766 

5813 

5860 

5907 

5954 

6001 

6048 

6095 

47 

5 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

6 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7oas 

7 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7454 

7501 

8 

7548 

7595 

764-2 

7688 

7735 

7782 

7829 

7875 

7922 

79G9 

9 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

930 

8483 

8530 

8576 

8623 

8670 

8716 

8763 

8810 

8856 

8903 

1 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

2 

9416 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

3 

9882 

9928 

9975 

0021 

0068 

0114 

0161 

0207 

0254 

0300 

4 

970347 

0393 

0440 

0486 

0533 

0579 

0626 

0672 

0719 

07'65 

5 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

6 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

7 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

2064 

2110 

2157 

8 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

9 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

940 

3128 

3174 

3220 

3266 

3313 

3359 

3405 

3451 

3497 

3543 

1 

3590 

3636 

3682 

3728 

3774 

3820 

3866 

3913 

3959 

4G05 

2 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

3 

4512 

4558 

4604 

4650 

4696  1 

4742 

4788 

4834 

4880 

4926 

4 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

46 

PROPORTIONAL  PARTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

47 
46 

4.7 
4.6 

9.4 
9.2 

14.1 
13.8 

18.8 
18.4 

£3  5 
23.0 

28.2 
27.6 

32.9 
32.2 

37.6 
36.8 

42  3 
41.4 

TABLE   XI.      LOGARITHMS   OF   NUMBERS. 


195 


[So.  945  L.  975.]                                 [No.  989  L.  995. 

N. 

0 

* 

i 

a 

4 

5 

6 

7 

8 

9 

Diff. 

945 

975432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

6 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

7 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

8 

6808 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

9 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

950 

7724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8135 

1 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

2 

8637  8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

3 

9093  i  9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

4 

9548  |  9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

5 

980003  |  0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

6 

(458  i  0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

7 

0912  0957  1003 

1048 

1093 

1139 

1184 

1229 

1275  1320 

8 

1366  1411   1456 

1501 

1547 

1592 

1637 

1683 

1728  1773 

9 

1819  j  1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

960 

2271  2316 

2362 

2407 

2452  ! 

2497 

2543 

2588 

2633 

2678 

1 

2723  !  2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

2 

3175 

3220 

3265 

3310 

3356  , 

3401 

3446 

3491 

3536 

3581 

3 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

4 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

5 

4527 

457'2 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

6 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

7 

5426 

5471 

5516 

5561 

5606 

5651 

5696 

5741 

5786 

5830 

8 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

9 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

970 

6772 

6817 

6861 

6906  6951 

6996 

7040 

7085 

7130 

7175 

1 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

2 

7666 

7711 

7756 

7800  j  7845 

7890 

7934 

7979 

8024 

8068 

3 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

4 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

5 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

6 

7 

9450 

9895 

9494 
9939 

9539 
9983 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

0028 

0072 

0117 

0161 

0206 

0250 

0294 

8 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

9 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

980 

1226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

1 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

2 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

3 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

4 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

5 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

6 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

7 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

44 

8 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

9 

5196 

5240 

5284 

5328 

5372 

5416 

5460 

5504 

5547 

5591 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

46    4.6 

9.2    13.8    18.4 

23.0 

27.6    32.2    36.8 

41.4 

45    4.5 

9.0    13.5    18.0 

22.5 

27.0    31.5    36.0 

40.5 

44    4.4 

8.8    13.2    17.6 

22.0 

26.4    30.8    35.2 

39.6 

43    4.3 

8.6    12.9    17.2 

21.5 

25.8    30.1    34.4 

38.7 

196 


TABLE   XI.      LOGARITHMS   OF   NUMBERS. 


No.  990  L.  995.] 

[No.  999  L.  999. 

N. 

990 
1 
2 
3 

4 
5 
6 
7 
8 
9 

0 

1 

2 

3 

4 

567 

8 

9 

Diff. 

995635 
6074 
6512 
6949 
7386 
7823 
8259 
8695 
9131 
9565 

5679 
6117 
6555 
6993 
7430 
7867 
8303 
8739 
9174 
9609 

5723 
6161 
6599 
7037 

7474 
7910 
8347 
8782 
9218 
9652 

5767 
6205 
6643 
7080 
7517 
7954 
8390 
8826 
9261 
9696 

5811 
6249 
6687 
7124 
7561 
7998 
8434 
8869 
9305 
9739 

5854     5898     5942 
6293     6337     6380 
6731     6774     6818 
7168     7212     7255 
7605     7648     7692 
8041     8085     8129 
8477     8521     8564 
8913     8956     9000 
9348     9392     9435 
9783     9826     9870 

5986 
6424 
6862 
7299 
7736 
8172 
8608 
9043 
9479 
9913 

6030 
6468 
6906 
7343 
7779 
8216 
8652 
9087 
9522 
9957 

44 

48 

CONSTANT  NUMBERS  AND  THEIR  LOGARITHMS. 

Symbol. 

Number. 

Logarithm. 

77 
277 
877 
477 

577 
677 
-777 
877 
977 

3.141  592653590 
6.283  185  307  180 
9.424  777  960  769 
12.566  370  614  359 
15.707963267950 
18.849555921  539 
21.991  148575  119 
25.132741228718 
28.274333882308 

0.497  149  872  694 
0.798179868358 
0.974271  127414 
1.099209864022 
1.196119877030 
1.275301  123078 
1.342247912708 
1.400239859686 
1.451392382133 

i71" 

i?7 

i77 

0 

0 

1 

4 

.523598775598 
.785  398  163  397 
.570  796  326  795 
.188790204786 

T.  71  8  998  622  310 
T-  895  089  881  366 
0.196119877030 
0.622  088  609  302 

H 

9 
31 

.869  604  401  089 
.006  276  680  293 

0.994  299  745  388 
1.491  449618082 

1//7 

1 

.772453850906 

0.248  574  936  347 

VTT 

1 

.464  591  887  562 

0.165716624231 

1/77 
180/77 
1/7T* 

1/1/77 

0.318309886184 
57.295779513025 
0.101321  183642 
0.564  189  583  548 
1.144729885849 

T.  502  850  127  306 
1.758  122632409 
T-005  700  254  612 
T-751  425063653 
0.058703021240 

arc  1° 
sin  1° 
arc  1' 
sin  1' 
arc  \" 
sin  1" 

0.017453292520 
0.017  452  406  417 
0.000  290  888  209 
0.000290888205 
0.000  004  848  137 
0.000  004  848  137 

"2".  241  877367591 
2~.241  855318418 
T.4637;;6  117207 
~£.  463  726  111  082 
•gr.685  574  866  824 
•5-.  685  574  866  822 

e 

1/Jf 

2 
0 
2 

.718  281  828  459 
.434  294  481  903 
.302  585  092  994 

0.434  294  481  903 
T.  637  784  31  1301 
0.362  215  688  699 

1/2 

1 

.414  313  562  373 

0.150514997832 

1/3 

1 

.732050807569 

0.238560627360 

** 

2 

.236  0(5 

7  977  477 

0.349  485  002  168 

TABLE  XII. 

LOGAKITHMIC  SINES,  COSINES,  TANGENTS, 
AND   COTANGENTS. 

Pages  198-242  give  values  of  these  functions  to  six  decimal  places  for 
every  minute  of  the  first  and  second  quadrants.  The  degrees  are  at  the 
top  and  bottom  of  the  pages  and  the  minutes  at  the  sides  below  or  above 
the  degrees.  For  example,  on  page  208,  the  angles  10°  26'  and  169°  34' 
have  log  sin  =9.257898,  while  79°  20'  and  100°  40'  have  log  cot  = 
9.274964. 

The  columns  headed  D.  1"  enable  interpolation  to  be  made  for 
seconds;  thus  for  10°  26'  15"  the  D.  1"  is  11.42  for  log  sin,  whence 
11.42X15=171  and  log  sin  for  this  angle  is  9.257898+171=9.258069. 
Also  for  163°  38'  15"  the  log  tan  is  9.467880-117=9.467763.  The 
computed  difference  is  to  be  added  or  subtracted  according  as  the 
tabular  values  of  the  function  increase  or  decrease  with  an  increase  in 
the  angle. 

The  columns  of  D.  1"  are  omitted  on  pages  198  and  199,  except  for 
log  cos;  while  other  columns  are  added  which  enable  intermediate 
values  of  the  other  functions  to  be  found  for  small  angles  more  accu- 
rately than  can  be  done  by  interpolation.  Thus  to  find  log  sin  A  and 
log  tan  A,  when  A  contains  seconds,  the  equations 

log  sin  A=S  +log  A ",  log  tan  A  =  T -flog  A", 

are  to  be  used,  A"  signifying  the  number  of  seconds  in  the  angle  A. 
For  example,  let  the  angle  A  be  1°  6'  33"  or  3993";  for  1°  6'  the  value 
of  S  is  taken  from  the  fourth  column  on  page  199  and  log  3993  from 
Table  XL  Then 

For  1°  6'  S  =4.685548 

log  3993  =3.601299 

log  sin  1°  6'  33"  =8.286847 
Similarly  for  0°  54'  12"  or  3252"  the  log  tan  is  found  as  follows: 

For  0°  54'          T  =4.685611 
log  3252  =3.512151 

log  tan  0°  54'  12"  =8.197762 

To  find  log  cot  for  a  small  angle  the  equation  log  cot  A  =C-log  A"  is 
to  be  used  where  C  is  taken  from  the  eighth  column.  For  example,  for 
1°  0'  16"  or  3616"  the  value  of  C  is  15.314381  and  that  of  log  3616  is 
3.558228,  whence  log  cot  1°  0'  16"  =11.756153. 

To  find  the  angle  from  a  given  logarithmic  function,  the  eye  must 
run  along  the  table  until  the  tabular  value  nearest  to  it  is  found. 
Thus,  when  log  tan  is  given  as  9.516910  this  is  found  on  page  216  and 
the  angle  is  either  18°  12'  or  161°  48'.  Again,  when  log  tan  is  given  as 
9.526004,  this  is  found  to  lie  between  9.525778  and  9.526197;  to  the 
first  value  corresponds  the  angle  18°  33'  and  the  D.  1"  is  6.98;  the 
difference  9.526004-9.525778  is  226  and  226/6.98=32.4",  so  that  the 
required  angle  is  18°  33'  32". 4. 

When  the  given  function  falls  on  page  198  or  199,  the  number  of 
seconds  is  found  by  the  equations 

log  A"  =log  sin  A  -S,     log  A"  =log  tan  A  -T,     log  A"  =C -log  cot  A. 

For  example,  given  log  tan  A  as  8.465371  for  which  T  is  4.685700; 
then  log  A"  =8.465371 -4. 685700  =3.779671  from  which  by  Table  XI 
there  is  found  A"  =6021",  and  hence  A  =1°  40'  21". 

W 


TABLE   XII.       LOGARITHMIC    SINES, 


II 

/ 

Sine. 

S   T 

Tang. 

Cotang. 

C 

Dl" 

Cosine. 

/ 

4.685 

15.314 

0 

0 

Inf.  neg. 

575 

575 

Inf.  neg. 

Inf.  pos. 

425 

ten 

60 

60 

1 

6.463726 

575 

575 

6.463726 

13.536274 

425 

ten 

59 

120 

2 

.764-156 

575 

575 

.764756 

.235244 

425 

ten 

58 

180 

3 

6.940847 

575 

575 

6.940847 

13.059153 

425 

ten 

57 

240 

4 

7.065786 

575 

575 

7.065786 

12.934214 

425 

ten 

56 

300 

5 

.162696 

575 

575 

.162696 

.837304 

425 

ten 

55 

300 

6 

.241877 

575 

575 

.241878 

.758122 

425 

.02 

9.999999 

54 

420 

7 

.308824 

575 

575 

.308825 

.691175 

425 

.00 

.999999 

53 

480 

8 

.366816 

574 

576 

.366817 

.633183 

424 

.00 

.999999 

52 

540 

9 

.417968 

574 

576 

.417970 

.582030 

424 

.00 

.999999 

51 

600 

10 

.463726 

574 

576 

.463727 

.536273 

424 

.02 

.999998 

50 

660 

11 

7.505118 

574 

576 

7.505120 

12.494880 

424 

.00 

9.999998 

49 

720 

12 

.542906 

574 

577 

.542909 

.457091 

423 

.02 

.999997 

48 

780 

13 

.577668 

574 

577 

.577672 

.422328 

423 

.00 

.999997 

47 

840 

14 

.609853 

574 

577 

.609857 

.390143 

423 

.02 

.999996 

46 

900 

15 

.639816 

573 

578 

.639820 

.360180 

422 

.00 

.999996 

45 

960 

16 

.667845 

573 

578 

.667849 

.332151 

422 

.02 

.91)9995 

44 

1020 

17 

.694173 

573 

578 

694179 

.30C821 

422 

.00 

.999995 

4b 

1080 

18 

.718997 

573 

579 

.719003 

.280997 

421 

.02 

.999994 

42 

1140 

19 

.742478 

573 

579 

.742484 

.257516 

421 

.02 

.999993 

41 

1200 

20 

.764754 

572 

j580 

.764761 

.235239 

420 

.00 

.999993 

40 

1260 

21 

7.785943 

572 

580 

7.785951 

12.214049 

420 

.C2 

9.999992 

3£ 

1320 

22 

.806146 

572 

!581 

.806155 

.  193845 

419 

.02 

.999991 

38 

1380 

23 

.825451 

572 

581 

.825460 

.  174540 

419 

.02 

.999990 

37 

1440 

24 

.843934 

571 

:582 

:  843944 

.156056 

418 

.02 

.999989 

36 

1500 

25 

.861662 

571 

;583 

.861674 

.138326 

417 

.00 

.999989 

35 

1560 

26 

.878695 

571 

583 

.878708 

.121292 

417 

.02 

.999988 

34 

162C 

27 

.895085 

570 

584 

.895099 

,104901 

416 

.02 

.999987 

33 

1680 

28 

.910879 

570 

584 

.910894 

.089106 

416 

.02 

.999986 

32 

1740 

29 

.926119 

570 

585 

.926134 

.073866 

415 

.02 

.999985 

31 

1800 

30 

.940842 

569 

586 

.940858 

.059142 

414 

.03 

.999983 

30 

1860 

31 

7.955082 

569 

587 

7.955100 

12.044900 

413 

.02 

9.999982 

29 

1920 

32 

.968870 

569 

587 

.968889 

.031111 

413 

.02 

.999981 

28 

1980 

33 

.982233 

568 

588 

.982253 

.017747 

412 

.02 

.999980 

27 

2040 

34 

7.995198 

568 

589 

7.995219 

12.004781 

411 

.02 

.999979 

26 

2100 

35 

8.007787 

567 

590 

8.007809 

11.992191 

410 

.03 

.9.9977 

25 

2160 

36 

.020021 

567 

591 

.020044 

.979956 

409 

.02 

.999976 

24 

2220 

37 

.031919 

566 

592 

.031945 

.968055 

408 

.02 

.999975 

23 

2280 

38 

.043501 

566 

593 

.043527 

.956473 

407 

.03 

.999973 

22 

2340 

39 

.054781 

566 

593 

.054809 

.945191 

407 

.02 

.999972 

21 

2400 

40 

.065776 

565 

594 

.065806 

.934194 

406 

t02 

.999971 

20 

8460 

41 

8.076500 

565 

595 

8.076531 

11.923469 

405 

.03 

9.999969 

19 

2520 

42 

.086965 

564 

596 

.086997 

.913003 

404 

.02 

.999968 

18 

2580 

43 

.097183 

564 

598 

.097217 

.902783 

402 

.03 

.999966 

17 

2640 

44 

.107167 

563 

599 

.107203 

.892797 

401 

.03 

.999964 

16 

2700 

45 

.116926 

562 

600 

.116963 

.883037 

400 

.02 

.999963 

15 

2760 

46 

.126471 

562 

601 

.126510 

.873490 

399 

,03 

.999961 

14 

2820 

47 

.135810 

561 

602 

.135851 

.864149 

398 

.03 

.999959 

13 

2880 

48 

.144953 

561 

603 

.144996 

.855004 

397 

.02 

.999958 

12 

2940 

49 

.153907 

560 

604 

.153952 

.846048 

396 

.03 

.999956 

11 

3000 

50 

.162681 

560 

605 

.162727 

.837273 

395 

.03 

.999954 

10 

3060 

51 

8.171280 

559 

607 

8.171328 

11.828672 

393 

.03 

9.999952 

9 

3120 

52 

.179713 

558  608 

.179763 

.820237 

392 

.03 

.999950 

8 

3180 

53 

.187985 

558!  609 

.188036 

.811964 

391 

.03 

.999948 

7 

3240 

54 

.196102 

557  611 

.196156 

.803844 

389 

.03 

.999946 

6 

3300 

55 

.204070 

556  1612 

.204126 

.795874 

388 

.03 

.999944 

5 

3?60 

56 

.211895 

556  i  '  613 

.211953. 

.788047 

387 

.03 

.999942 

4 

3120 

57 

.219581 

555  i  1  615 

.219641 

.780359 

385 

.03 

.999940 

•3 

3180 

58 

.227134 

554  j  616 

.227195 

.772805 

384 

.03 

.999938 

2 

i  3540 

59 

.234557 

554  !  618 

.234621 

.765379 

382 

.03 

.999936 

1 

|3600 

60 

8.241855 

553  i  619 

8.241921 

11.758079 

381 

.03 

9.999934 

0 

4.6S5 

15.314 

/« 

/ 

Cosine. 

Cotang. 

Tang. 

Fr 

Sine. 

> 

COSINES,   TANGENTS,    AND   COTANGENTS. 


178* 


// 

/ 

Sine. 

S  T 

Tang. 

Cotang. 

C 

Dl" 

Cosine. 

/ 

4.685 

15.314 

3600 

0 

8.241855 

553 

619 

8.241921 

11.758079 

381 

9.999934 

60 

3660 

1 

.249033 

552 

620 

.249102 

.750898 

380 

.03 

.999932 

59 

3720 

2 

.256094 

551 

622 

.256165 

.743835 

378 

•95   .999929 

58 

3780 

3 

.263042 

551 

623 

.263115 

.736885 

377 

'no   .999927 

57 

3840 

4 

.269881 

550 

625 

.269956 

.730044 

375 

.Uo  1 

.999925 

56 

3900 

5 

.276614 

549 

627 

.276691 

.723309 

373 

.05 

no 

.999922 

55 

3960 

6 

.283243 

548 

628 

.283323 

.716677 

372 

.Uu 

.999920 

54 

4020 

7 

.289773 

547 

630 

.289856 

.710144 

370 

.03 

.999918 

53 

4080 

8 

.296207 

546 

632 

.296292 

.703708 

368 

.05 

.999915 

52 

4140 

9 

.302546 

546 

633 

.302634 

.697366 

367 

.03 

.999913 

51 

4200 

10 

.308794 

545 

635 

.308884 

.691116 

365 

.05 

.999910 

50 

4260  !  11 

8.314954 

544 

637 

8.315046 

11.684954 

363 

.05 

9.999907 

49 

4320  12 

.321027 

543 

638 

.321122 

.67'8878 

362 

.03 

.999905 

48 

4380  13 

.327016 

542 

640 

.327114 

.672886 

360 

.05 

.999902 

47 

4440  14 

.332924 

541 

642 

.333025 

.666975 

358 

•9o   .999899 

46 

4500  15 

.338753 

540 

644 

.338856 

.661144 

356 

.03 

.999897 

45 

4560  16 

.344504 

539 

646 

.344610 

.655390 

354 

.05 

.999894 

44 

4620  17 

.350181 

539 

648 

.350289 

.649711 

352 

.05 

.999891 

43 

4680 

18 

.355783 

538 

649 

.355895 

.644105 

351 

.05 

.999888 

42 

4740 

19 

.361315 

537 

651 

.361430 

.638570 

349 

•S5   .999885 

41 

4800 

20 

.366777 

536 

653 

.366895 

.633105 

347 

*U5   .999882 

40 

4860 

21 

8.372171 

535 

655 

8.372292 

11.627708 

345 

.05 

(\K 

9.999879 

39 

4920 

22 

.377499 

534 

657 

.377622 

.622378 

343 

.UO 
ne 

.999876 

38 

4980 

23 

.382762 

533 

659 

.382889 

.617111 

341 

.UO 

.999873 

37 

5040 

24 

.887962 

532 

661 

.388092 

.611908 

339 

.05 

(\K 

.999870 

36 

5100 

25 

.393101 

531 

663 

.393234 

.606766 

337 

.UO 

OK 

.999867 

35 

5160 

26 

.398179 

530 

666 

.398315 

.601685 

334 

.UO 

OK 

.999864 

34 

5220 

27 

.403199 

529 

668 

.403338 

.596662 

332 

,uo 

.999861 

33 

5280 

28 

.408161 

527 

670 

.408304 

.591696 

330 

.05 

.999858 

32 

5340 

29 

.413068 

526 

672 

.413213 

.586787 

328 

.07 

OK 

.999854 

31 

5400 

30 

.417919 

525 

674 

.418068 

.581932 

326 

.UO 

.999851 

30 

5460 

31 

8.422717 

524 

676 

8.422869 

11.577131 

324 

.05 

CY7 

9.999848 

29 

5520 

32 

.427462 

523 

679 

.427618 

.572382 

321 

.U< 

OK 

.999844 

28 

5580 

33 

.432156 

522 

681 

.432315 

.567685 

319 

.UO 

OK 

.999841 

27 

5640 

34 

.436800 

521 

683 

.436962 

.563038 

317 

.UO 

.999838 

26 

5700 

35 

.441394 

520 

685 

.441560 

.558440 

315 

.07 

C\f\ 

.999834 

25 

5760 

36 

.445941 

518 

688 

.446110 

.553890 

312 

.UO 

.999831 

24 

5820 

37 

.450440 

517 

690 

.450613 

.549387 

310 

.07 

C\f\ 

.999827 

23 

5880 

38 

.4E4893 

516 

693 

.455070 

.544930 

307 

.UO 
A7 

.999824 

22 

5940 

39 

.459301 

515 

695 

.459481 

.540519 

305 

.u< 

O7 

.999820 

21 

6000 

40 

.463665 

514 

697 

.463849 

.536151 

303 

»vi 

.999816 

20 

6060 

41 

8.467985 

512 

700 

8.468172 

11.531828 

300 

.05 

O7 

9.999813 

19 

6120 

42 

.472263 

511 

702 

.472454 

.527546 

298 

.u< 

07 

.999809 

18 

6180 

43 

.476498 

510 

705 

.476693 

.523307 

295 

.u< 

O7 

.999805 

17 

6240 

44 

.480693 

509 

707 

.480892 

.519108 

293 

.U< 

07 

.999801 

16 

6300 

45 

.484848 

507 

710 

.485050 

.514950 

290 

.Ui 
OK 

.919797 

15 

6360 

46 

.488963 

506 

713 

.489170 

.510830 

287 

.UO 

nr< 

.999794 

14 

6420 

47 

.493040 

505 

715 

.493250 

.506750 

285 

,V( 

f!7 

.999790 

13 

6480 

48 

.497078 

503 

718 

.497293 

.502707 

282 

.U< 
CV7 

.999786 

12 

6540 

49 

.501080 

502 

720 

.501298 

.498702 

280 

.u< 

O7 

.999782 

11 

6600 

50 

.505045 

501 

723 

.505267 

.494733 

277 

.u< 

.999778 

10 

6660 

51 

8.508974 

499 

726 

8.509200 

11.490800 

274 

.07 

no 

9.999774 

9 

6720 

52 

.512867 

498 

729 

.513098 

.486902 

271 

.UO 
fl7 

.999769 

8 

6780 

53 

.516726 

497 

731 

.516961 

.483039 

269 

.Ui 

O7 

.999765 

7 

6840 

54 

.520551 

495 

734 

.520790 

.479210 

266 

.U< 

O7 

.999761 

6 

6900 

55 

.524343 

494 

737 

524586 

.475414 

263 

.Ui 
fi7 

.999757 

5 

6960 

56 

.528102 

492 

740 

.528349 

.471651 

260 

.U< 

no 

.999753 

4 

7020 

57 

.531828 

491 

743 

.532080 

.467920 

257 

.Uo 
07 

.999748 

3 

7080 

58 

.535523 

490 

745 

.535779 

.464221 

255 

.Ui 

O7 

.999744 

2 

7140 

59 

.539186 

488 

748 

.539447 

.460553 

252 

,U< 

no 

.999740 

1 

7200 

60 

8.542819 

487 

751 

8.543084 

11.456916 

249 

•UO 

9.999735 

0 

4.685 

15.314 

n 

/ 

Cosine. 

Cotang. 

Tang. 

! 

Dl* 

Sine. 

; 

TABLE  XII.      LOGARITHMIC   SI1STES, 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.  1'. 

Cotang. 

' 

0 

1 

\ 

i 
i 

8  542819 
.546422 
.549995 
.553539 
.557054 
.560540 
.563999 
.567431 

60.05 
59.55 
59.07 
58.58 
58.10 
57.65 
57.20 

!  9.999735 
.999731 
.999726 
.999722 
.999717 
.999713 
.999708 
.999704 

.07 
.08 
.07 
.08 
.07 
.08 
.07 

AQ 

8.5430S4 
.546691 
.550268 
.553817 
.557336 
.560828 
.564291 
.567727 

60.12 
59.62 
59.15 
58.65 
58.20 
57.72 
57.27 

11.456916 
.453309 
.449732 
.446183 
.442664 
.439172 
.435709 
.432273 

60 
59 
58 
57 
56 
55 
54 
53 

8 
9 
10 

.570836 
.574214 
.577566 

56.  7£ 
56.30 
55.87 
55.43 

.999699 
.999694 
.999689 

.(JO 

.08 
.08 
.07 

.571137 
.574520 

.577877 

56.  83 
56.38 
55.95 
55.52 

.428863 
.425480 
.422123 

52 
51 
50 

11 
12 
13 
14 
15 
16 
17 

8.580892 
.584193 
.587469 
.590721 
.593948 
.597152 
.600332 

55.02 
54.60 
54.20 
53.78 
53.40 
53.00 

9.999685 
.999680 
.999675 
.999670 
.999665 
.999660 
.999655 

.08 
.08 
.08 
.08 
.08 
.08 

8.581208 
.584514 
.587795 
.591051 
.594283 
.597492 
.600677 

55.10 
54.68 
54.27 
53.87 
53.48 
53.08 

11.418792 
.415486 
.412205 
.408949 
.405717 
.402508 
.399323 

49 
48 
47 
46 
45 
44 
43 

18 

.603489 

52.62 

.999650 

.08 

.603S39 

52.70 

.396161 

42 

19 

.606623 

52.23 

.999645 

.08 

.606978 

52.32 

.393022 

41 

20 

.609734 

51.85 

.999640 

.08 

.610094 

51.93 

.389906 

40 

51.48 

.08 

51.58 

21 
22 

23 
24 
25 
26 
27 
28 

8.612823 
.615891 
.618937 
.621962 
.624965 
.627948 
.630911 
.633854 

51.13 
50.77 
50.42 
50.05 
49.72 
49.38 
49.05 

9.999635 
.999629 
.999624 
.999619 
.999614 
.999608 
.999603 
.999597 

.10 
.08 
.08 
.08 
.10 
.08 
.10 

AQ 

8.613189 
.616262 
.619313 
.622343 
.625352 
.628340 
.631308 
.634256 

51.22 
50.85 
50.50 
50.15 
49.80 
49.47 
49.13 

11.386811 
.383738 
.380687 
.377657 
.374648 
.371660 
.368692 
.365744 

39 

38 
37 
36 
35 
34 
33 
32 

29 
30 

.636776 
.639680 

48.70 
48.40 
48.05 

.999592 
.999586 

,uo 
.10 
.08 

.637184 
.640093 

4S!48 
48.15 

.362816 
.359907 

31 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

8.642563 
.645428 
.648274 
.651102 
.653911 
.656702 
.659475 
.662230 
.664968 
.667689 

47.75 
47.43 
47.13 
46.82 
46.52 
46.22 
45.92 
45.63 
45.35 

9.999581 
1  .999575 
.999570 
.999564 
.999558 
.999553 
.999547 
.999541 
.999535 
.999529 

.10 
.08 
.10 
.10 
.08 
.10 
.10 
'  .10 
.10 

8.642982 
.645853 
.648704 
.651537 
.654352 
.657149 
.659928 
.662689 
.665433 
.668160 

47.85 
47.52 
47.22 
46.92 
46.62 
46.32 
46.02 
45.73 
45.45 

11.357018 
.354147 
.351296 
.348463 
.345648 
.342851 
.340072 
.337311 
.334567 
.331840 

29 
2S 
27 
26 
25 
24 
23 
22 
21 
20 

45.07 

.08 

45.17 

41 
42 

43 
44 
45 
46 

47 

8.670393 
.673080 
.675751 
.678405 
.681043 
.683665 
.686272 

44.78 
44.52 
44.23 
43.97 
43.70 
43.45 

9.999524 
.999518 
.999512 
.999506 
.999500 
.999493 
.999487 

.10 
,  .10 
.10 
.10 
.12 
.10 

8.670870 
.673563 
.676239 
.678900 
.681544 
.684172 
.686784 

44.88 
44.60 
44.35 
44.07 
43.80 
43.53 

11.329130 
.326437 
.323761 
.321100 
.318456 
.315828 
.313216 

19 
18 
17 
16 
15 
14 
13 

48 
49 
50 

.688863 
.6914M8 
.693998 

43.18 
42.92 
42.67 
42.42 

.999481 
.999475 
.999469 

.10 
.10 
.10 
.10 

.689381 
.691963 
.694529 

43.28 
43.03 
42.77 
42.53 

.310619 
.308037 
.305471 

12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

8.696543 
.699073 
.701589 
.704090 
.706577 
.709049 
.711507 
.713952 
.716383 
8.718800 

42.17 
41.93 
41.68 
41.45 
41.20 
40.97 
40.75 
40.52 
40.28 

9.999463 
.999456 
.999450 
.999443 
.999437 
.999431 
.999424 
.999418 
.999411 
9.999404 

.12 
.10 
.12 
.10 
.10 
.12 
.10 
.12 
.12 

8.697081 
.699617 
.702139 
.704646 
.707140 
.709618 
.712083 
.714534 
.716972 
8.719396 

42.27 
42.03 
41.78 
41.57 
41.30 
41.08 
40.85 
40.63 
40.40 

11.302919 
.300383 
.297861 
.-295354 
.292860 
.290382 
.287917 
.285466 
.283028 
11.280604 

9 
8 
7 
6 
5 
4 
8 
2 
1 
0 

' 

Cosine. 

D  r. 

Sine.    D.  1". 

Cotang.  i  D.  1*.  1  Tang. 

' 

COSINES,    TANGENTS,,    AND   COTANGENTS. 


' 

Sine. 

D.  1". 

Cosine.   D.  1". 

Tang. 

D.r. 

Cotang. 

' 

0 

1 

8.718800 
.721204 

40.07 

qQ  OK 

9.999404 
.999398 

.10 

8.719396 
.721806 

40.17 

qQ  QW 

11.280604 
.278194 

60 
59 

2 

.723595 

oy.oo 

.999391 

19 

.724204 

oy  .v  ( 

.275796 

58 

3 

4 
5 
6 
7 

.725972 
.728337 
.730688 
.733027 
.735354 

39.'42 
39.18 
38.98 
38.78 

.999384 
.999378 
.999371 
.999364 
.999357 

tiX 

.10 
.12 
.12 
.12 

.726588 
.728959 
.731317 
.733663 
.735996 

39!  52 
39.30 
39.10 

38.88 

qo  (*O 

.273412 
.271041 
.268683 
.266337 
.264004 

57 
56 
55 
54 
53 

8 
9 

.737667 
.739969 

38.55 
3837 

.999350 
.999343 

.12 
.12 

.738317 
.740626 

OO.OO 

38.48 

OQ  07 

.261683 
.259374 

52 
51 

10 

.742259 

38.17 
37.95 

.999336 

.12 
.12 

.742922 

38^08 

.257078 

50 

11 
12 
13 
14 
15 
16 
17 
18 
19 

8.744536 
.746802 
.749055 
.751297 
.753528 
.755747 
.757955 
.760151 
.762337 

37.77 
37.55 
37.37 
37.18 
36.98 
36.80 
36.60 
36.43 
36  2S 

9.999329 
.999322 
.999315 
.999308 
.999301 
.999294 
.999287 
.999279 
.999272 

.12 
.12 
.12 
.12 
.12 
.12 
.13 
.12 

8.745207 
.747479 
.749740 
.751989 
.754227 
.756453 
.758668 
.760872 
.763065 

37.87 
37.68 
37.48 
37.30 
37.10 
•  36.92 
36.73 
36.55 
36  35 

11.254793 
.252521 
.250260 
.248011 
.245773 
.243547 
.241332 
.239128 
.236935 

49 
48 
47 
46 
45 
44 
43 
42 
41 

20 

.764511 

36^07 

.999265 

!l3 

.765246 

36^18 

.234754 

40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

8.766675 
.768828 
.770970 
.773101 
.775223 
.777333 
.779434 
.781524 
.783605 
.785675 

35.88 
35.70 
35.52 
35.37 
35.17 
35.02 
34.83 
34.68 
34.50 
34.35 

9.999257 
.999250 
.999242 
.999235 
.999227 
.999220 
.999212 
.999205 
.999197 
.999189 

.12 
.13 
.12 
.13 
.12 
.13 
.12 
.13 
.13 
.13 

8.767417 
.769578 
.771727 
.773866 
.775995 
.778114 
.780222 
.782320 
.784408 
.786486 

36.02 
35.82 
35.65 
35.48 
35.32 
35.13 
34.97 
34.80 
34.63 
34.47 

11.232583 
.230422 
.228273 
.226134 
.224005 
.221886 
.219778 
.217680 
.215592 
.213514 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 

8.787736 
.789787 
.791828 

34.18 
34.02 

qq  OR 

9.999181 
.999174 
.999166 

.12 
.13 

8.788554 
.790613 
.792662 

34.32 
34.15 

OO  QQ 

11.211446 
.209387 
.207338 

29 

28 
27 

34 

.793859 

OO.  OO 

qq  r-A 

.999158 

.13 

1°. 

.794701 

oo.  9o 

qq  QO 

.205299 

26 

36 
37 

38 

.795881 
.797894 
.799897 
.801892 

oo  .  (U 

33.55 
33.38 
33.25 

.999150 
.999142 
.999134 
.999126 

.lo 

.13 
.13 
.13 

.796731 
.798752 
.800763 
.802765 

Oo.oo 

33.68 
33.52 
33.37 

qq  on 

.203269 
.201248 
.199237 
.197235 

25 
24 
23 
22 

39 
40 

.803876 
.805852 

33.07 
32.93 
32.78 

.999118 
.999110 

.13 
.13 
.13 

.804758 
.806742 

OO.JW 

33.07 
32.92 

.195242 
.193258 

21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 

8.807819 
.809777 
.811726 
.813667 
.815599 
.817522 
.819436 
.821343 
.823240 

32.63 
32.48 
32.35 
32.20 
32.05 
31.90 
31.78 
31.62 
q-<  CA 

9.999102 
.999094 
.•999086 
.999077 
.999069 
.999061 
.999053 
.999044 
.999036 

.13 
.13 
.15 
.13 
.13 
.13 
.15 
.13 

•JK 

8.808717 
.810683 
.812641 
.814589 
.816529 
.818461 
.820384 
.822298 
.824205 

32.77 
32.63 
32.47 
32.33 
32.20 
32.05 
31.90 
31.78 
31  63 

11.191283 
.189317 
.187359 
.185411 
.183471 
.181539 
.179616 
.177702 
.175795 

19 
18 
17 
16 
15 
14 
13 
12 
11 

50 

.825130 

ol  .OU 

31.35 

.999027 

.  1O 

.13 

.826103 

31  .'48 

.173897 

10 

51 
52 
53 

54 
55 
56 

57 

8.827011 
.828884 
.830749 
.832607 
.834456 
.836297 
.838130 

31.22 
31.08 
30.97 
30.82 
30.68 
30.55 

9.999019 
.999010 
.999002 
.998993 
.998984 
.998976 
.998967 

.15 
.13 
.15 
.15 
.13 
.15 

8.827992 
.829874 
.831748 
.833613 
.835471 
.837321 
.839163 

31.37 
31.23 
31.08 
30.97 
30.83 
30.70 

11.172008 
.170126 
.168252 
.166387 
.164529 
.162679 
.160837 

9 
8 
7 
6 
5 
4 
3 

58 
£9 
CO 

.839956 
.841774 

8.843585 

30.43 
30.30 
30.18 

.998958 
.998950 
9.998941 

.15 
.13 

.840998 
.842825 
8.844644 

80.'  45 
30.32 

.159002 
.157175 
11.155356 

2 

'  \  Cosine. 

D  1". 

Sine. 

D.  r. 

Cotang. 

D.I". 

Tang. 

' 

o.m 


TABLE   XII.       LOGARITHMIC    SINES, 


! 

' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

i 

3 

4 
5 
6 

7 
8 
9 

8.843585 
.845387 
.847183 
.848971 
.850751 
.852526 
.854291 
.856049 
.857801 
.859546 

30.03 
29.93 
29.80 
29.67 
29.57 
29.43 
29.30 
29.20 
29.08 

9.998941 
.998932 
.998923 
.998914 
.998905 
.998896 
.998887 
.998878 
.998869 
.998860 

.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
1  f\ 

8.844644 
.846455 
.848260 
.850057 
.851846 
.853628 
.855403 
.857171 
.858932 
.800686 

30.18 
30.08 
29.95 
29.82 
29.70 
29.58 
29.47 
29.35 
29.23 

90  -<9 

11.155356 
.153545 
.151740 
.149943 
.148154 
.146372 
.144597 
.142829 
.141068 
.139314 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 

.861283 

28^85 

.998851 

.17 

.862433 

29.00 

.137567 

50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

8.863014 
.864738 
.866455 
.868165 
.869868 
.871565 
.873255 
.874938 
.876615 
.878285 

28.73 
28.62 
28.50 
28.38 
28.28 
28.17 
28.05 
27.95 
27.83 
27.73 

9.998841 
.998832 
.998823 
.998813 
.998804 
.998795 
.998785 
.998776 
.998766 
.998757 

.15 
.15 
.17 
.15 
.15 
.17 
.15 
.17 
.15 
.17 

8.864173 
.865906 
.867632 
.869351 
.871064 
.872770 
.874469 
.876162 
.877849 
.879529 

28.88 
28.77 
28.65 
28.55 
28.43 
28.32 
28.22 
28.12 
28.00 
27.88 

11.135827 
.134094 
.  132368 
.130649 
.128936 
.127230 
.125531 
.123838 
.122151 
.120471 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

8.879949 
.881607 
.883258 
.884903 
.886542 
.888174 
.889801 
.891421 
.893035 
.894643 

27.63 
27.52 
27.42 
27.32 
27.20 
27.12 
27.00 
26.90 
26.80 
26.72 

9.998747 
.998738 
.998728 
.998718 
.998708 
.998699 
.998689 
.998679 
.998669 
.998659 

.15 
.17 
.17 
.17 
,15 
17 
17 
.17 
.IT 

.ir 

8.881202 
.882869 
.884530 
.886185 
.887833 
.889476 
.891112 
.892742 
.894366 
.895984 

27.78 
27.68 
27.58 
27.47 
27.38 
27.27 
27.17 
27.07 
26.97 
26.87 

11.118798 
.117131 
.115470 
.113815 
.112167 
.110524 
.108888 
.107258 
.105634 
.104016 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

8.896246 
.897842 
.899432 
.901017 
.902596 
.904169 
.905736 
.907297 
.908853 
.910404 

26.60 
26.50 
26.42 
26.32 
26.22 
26.12 
26.02 
25.93 
25.85 
25.75 

9.998649 
.998639 
.998629 
.998619 
.998609 
.998599 
.998589 
.998578 
.998568 
.998558 

.17 
.17 
.17 
.17 
.17 
.17 
.18 
.17 
.17 
.17 

8.897596 
.899203 
.900803 
.902398 
.903987 
.905570 
.907147 
.908719 
.910285 
.911846 

26.78 
26.67 
26.58 
26.48 
26.38 
26.28 
26.20 
26.10 
26.02 
25.92 

11.102404 
.100797 
.099197 
.097602 
.096013 
.094430 
.092853 
.091281 
.089715 
.088154 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 

43 
44 
45 
46 

47 
48 
49 
50 

8.911949 
.913488 
.915022 
.916550 
.918073 
.919591 
.921103 
.922610 
.924112 
.925609 

25.65 
25.57 
25.47 
25.38 
25.30 
25.20 
25.12 
25.03 
24.95 
24.85 

9.998548 
.998537 
.998527 
.998516 
.998506 
.998495 
.998485 
.998474 
.998464 
.998453 

.18 
.17 
.18 
.17 
.18 
.17 
.18 
.17 
.18 
.18 

8.913401 
.914951 
.916495 
.918034 
.919568 
.921096 
.922619 
.924136 
.925649 
.927156 

25.83 
25.73 
25.63 
25.57 
25.47 
25.38 
25.28 
25.22 
25.12 
25.03 

11.086599 
.085049 
.083505 
.081966 
.080432 
.078904 
.077381 
.075864 
.074351 
.072844 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
55 

8.927100 
.928587 
.930068 
.931544 
.933015 

24.78 
24.68 
24.60 
24.52 

9.998442 
.998431 
.998421 
.998410 
.998399 

.18 
.17 
.18 
.18 

1Q 

8.928658 
.930155 
.931647 
.933134 
.934616 

24.95 

24.87 
24.78 
24.70 

11.071342 
.069845 
.068353 
.066866 
.065384 

9 
8 
7 
6 
5 

56 
57 
58 
59 
60 

.934481 
.935942 
.937398 
.938850 
8.940296 

24.35 
24.27 
24.20 
24.10 

.998388 
.998377 
.998366 
.998355 
9.998344 

.18 
.18 
.18 
.18 

.936093 
.937565 
.939032 
.940494 
8.941952 

24.53 
24.45 
24.37 
24.30 

.063907 
.062435 
.060968 
.059506 
11.058048 

4 

3 
2 
1 
0 

/ 

Cosine. 

D.  1". 

Sine. 

D.  1". 

Cotang.  D.  1". 

Tang. 

' 

94° 


COSINES,    TANGENTS,    AND    COTANGENTS. 


/ 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 
1 

2 
3 

8.940296 
.941738 
.943174 
.944606 

24.03 
23.93 

23.87 

9.998344 
.998333 
.998322 
.998311 

.18 
.18 
.18 

18 

8.941952 
.943404 
.944852 
.946295 

24.20 
24.13 
24.05 

11.058048 
.056596 
.055148 
.053705 

60 
59 
58 
57 

4 
5 

.946034 
.947456 

23.  80 
23.70 

.998300 
.998289 

.18 

.18 

.947734 
.949168 

23^90 

.052266 
.050832 

56 
55 

6 

7 

.948874 
.950287 

23!  55 

.998277 
.998266 

!l8 

•f  Q 

.950597 
.952021 

23.82 
23.73 

.049403 
.047979 

54 
53 

8 
9 
10 

.951696 
.953100 
.954499 

23^40 
23.32 
23.25 

.998255 
.998243 
.998232 

.  lo 

.20 
.18 
.20 

.953441 
.954856 
.956267 

23.67 
23.58 
23.52 
23.45 

.046559 
.045144 
.043733 

52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

8.955894 
.957284 
.958670 
.960052 
.961429 
.962801 
.964170 
.965534 
.966893 
.968249 

23.17 
23.10 
23.03 
22.95 
22.87 
22.82 
22.73 
22.65 
22.60 
22.52 

9.998220 
.998209 
.998197 
.998186 
.998174 
.998163 
.998151 
.998139 
.998128 
.998116 

.18 
.20 
.18 
.20 
.18 
.20 
.20 
.18 
.20 
.20 

8.957674 
.959075 
.960^3 
.961866 
.963255 
.964639 
.966019 
.967394 
.968766 
.970133 

23.35 
23.30 
23.22 
23.15 
23.07 
23.00 
22.92 
22.87 
22.78 
22.72 

11.042326 
.040925 
.039527 
.038134 
.036745 
.035361 
.033981 
.032606 
.031234 
.029867 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

8.969600 
.970947 
.972289 
.973628 
.974962 
.976293 
.977619 
.978941 
.980259 
.981573 

22.45 
22.37 
22.32 
22.23 
22.18 
22.10 
22.03 
21.97 
21.90 
21.83 

9.998104 

.998092 
.998080 
.998068 
.998056 
.998044 
.998032 
.998020 
.998008 
.997996 

.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
,20 

8.971496 
.972855 
.974209 
.975560 
.976906 
.978248 
.979586 
.980921 
.982251 
.983577 

22.65 
22.57 
22.52 
22.43 
22.37 
22.30 
22.25 
22.17 
22.10 
22.03 

11.028504 
.027145 
.025791 
.024440 
.023094 
.021752 
.020414 
.019079 
.017749 
.016423 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 

8.982883 
.984189 
.985491 
.986789 

21.77 
21.72 
21.63 

91  *V7 

9.997984 
.997972 
.997959 
.997947 

.20 
.22 

.20 

20 

8.984899 
.986217 
.987532 

.988842 

21.97 
21.92 
21.83 

91  7ft 

11.015101 
.013783 
.012468 
.011158 

29 

28 
27 
26 

35 
36 

37 

.988083 
.989374 
.990660 

Al  .  O< 

21.52 
21.43 

.997935 
.997922 
.997910 

!22 

.20 

.990149 
.991451 
.992750 

Al  .  (Q 

21.70 
21.65 

91  P»ft 

.009851 
.008549 
.007250 

25 
24 
23 

38 

.991943 

21.38 

91  ^9 

.997897 

*20 

.994045 

Al  .  Oo 
91  ^ 

.005955 

22 

39 
40 

.993222 
.994497 

Al  .  oA 

21.25 
21.18 

.997'885 
.997872 

.'22 
.20 

.995337 
.996624 

Al  .Oo 

21.45 
21.40 

.004663 
.003376 

21 
20 

41 
42 
43 
44 

8.995768 
.997036 
.998299 
8.999560 

21.13 
21.05 

21.02 

9.997860 
.997847 
.997835 
.997822 

.22 
.20 
.22 

8.997908 
8.999188 
9.000465 
.001738 

.  21.33 
21.28 
21.22 

11.002092 
11.000812 
10.999535 

.998262 

19 
18 
17 
16 

45 

9.000816 

oft  ca 

.997809 

9ft 

.003007 

91  'ftP 

.996993 

15 

46 

.002069 

9ft  89 

.997797 

*O9 

.004272 

Al  .Uo 

.995728 

14 

47 
48 
49 
50 

.003318 
.004563 
.005805 
.007044 

A\)  .  o/* 

20.75 
20.70 
20.65 
20.57 

.997784 
.997771 
.997758 
.997745 

]22 

.22 
.22 

.22 

.005534 
.006792 
.008047 
.009298 

20^97 
20.92 
20.85 
20.80 

.994466 
.993208 
.991953 
.990702 

13 

12 
11 
10 

51 

9.008278 

9.997732 

99 

9.010546 

9ft  73 

10.989454 

9 

52 
53 
54 
55 
56 
57 
58 
59 
60 

.009510 
.010737 
.011962 
.013182 
.014400 
.015613 
.016824 
.018031 
9.019235 

20^45 
20.42 
20.33 
20.30 
20.22 
20.18 
20.12 
20.07 

.997719 
.997706 
.997693 
.997680 
.997667 
.997654 
.997641 
.997628 
9.997614 

!22 
.22 
.22 
.22 

.22 
.22 
.22 
.23 

.011790 
.013031 
.014268 
.015502 
.016732 
.017959 
.019183 
.020403 
9.021620 

t&J  .  id 

20.68 
20.62 
20.57 
20.50 
20.45 
20.40 
20.33 
20.28 

.988210 
.986969 
.985732 
.984498 
.983268 
.982041 
.980817 
.979597 
10.978380 

8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  1'.  1   Sine. 

D.1-. 

Cotang. 

D.  1". 

Tang. 

1 

TABLE   XII.      LOGARITHMIC   SINES, 


1 

' 

Sine. 

D.  1", 

Cosine. 

D.  r. 

Tang. 

D.  1'. 

Cotang. 

' 

0 

1 

2 

9.019235 
.020435 
.021632 

20.00 
19.95 

9.997614 
.997601 
.997588 

.22 
.22 

oq 

9.021620 
.022834 
.024044 

20.23 
20.17 

10.978380 
.977166 
.975956 

60 

59 

58 

3 
4 
5 
6 

.022825 
.024016 
.025203 
.026386 

19.88 
19.85 
19.78 
19.72 

1Q  fift 

.997'574 
.997561 
.997547 
.997534 

,«*o 

.22 
.23 
.22 

OQ 

.025251 
.026455 
.027655 

.028852 

20.12 
20.07 
20.00 
19.95 

1Q  Qrt 

.974749 
.973545 
.972:345 
.971148 

57 
56 
55 

54 

8 
9 
10 

.027567 
.028744 
.029918 
.031089 

iy  .00 
19.62 
19.57 
19  52 
19.47 

.997520 
.997507 
.997493 
.997480 

.280 

.22 

.23 
.22 
.23 

.030046 
.031237 
.032425 
.033609 

iy  .yu 
19.85 
19.80 
19.73 
19.70 

.969954 
.968763 
.967575 
.966391 

53 

52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.032257 
.033421 
.034582 
.035741 
.036896 
.038048 
.039197 
.040342 
.041485 
.042625 

19.40 
19.35 
19.32 
19.25 
19.20 
19.15 
19.08 
19.05 
19.00 
18.95 

9.997466 
.997452 
.997439 
.997425 
.997411 
.997397 
.997383 
.997369 
.997355 
.997341 

.23 
.22 
.23 
.23 
.23 
.23 
.23 
.23 
.23 
.23 

9.034791 
.035969 
.037144 
.038316 
.039485 
.040651 
.041813 
.042973 
.044130 
.045284 

19.63 
19.58 
19.53 
19.48 
19.43 
19.37 
19.33 
19.28 
19.23 
19.17 

10.965209 
.964031 
.962856 
.961684 
.960515 
.959349 
.958187 
.957027 
.955870 
.954716 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 

9.043762 
.044895 

18.88 

-(0  OK 

9.997327 
.997313 

.23 

9.046434 
.047582 

19.13 

IQ  nft 

10.953566 
.952418 

39 

38 

23 

.046026 

lO.oO 

1R  Art 

.997299 

oq 

.048727 

iy  .uo 
10  fvi 

.951273 

37 

24 
25 
26 
27 

28 
29 
30 

.047154 
.048279 
.049400 
.050519 
.051635 
.052749 
.053859 

lo.ou 

18.75 
18.68 
18.65 
18.60 
18.57 
18.50 
18.45 

.997285 
.997271 
.  99725  7 
.997242 
.997228 
.997214 
.997199 

.30 

.23 
.23 
.25 
.23 
.23 
.25 
.23 

.049869 
.051008 
.052144 
.053277 
.054407 
.055535 
.056659 

iy  .uo 

18.98 
18.93 
18.88 
18  83 
18.80 
18.73 
18.70 

.950131 
.948992 
.947856 
.946723 
.945593 
.944465 
.943341 

36 
35 
34 
33 
32 
31 
30 

31 
32 

9.054966 

.056071 

18.42 

-fo  qK 

9.997185 
.997170 

.25 

9q 

9.057781 

.058900 

18.65 

1R  fiO 

10.942219 
.941100 

29 

28 

33 
34 
85 
36 
37 
38 

.057172 
.058271 
.059367 
.060460 
.061551 
.062639 

lo.oO 

18.32 
18.27 
18.SJ2 

18.18 
18.13 

18  08 

.997156 
.997141 
.997127 
.997112 
.997098 
.997083 

,4iO 

.25 
.23 
.25 
.2c 
.25 

OK 

.060016 
.061130 
.062240 
.063348 
.064453 
.065556 

lo.ou 
18.57 
18".  50 
18.47 
18.42 
18.38 
is  9.9 

.939984 
.938870 
.937760 
.936652 
.935547 
.934444 

27 
26 
25 
24 
23 
22 

39 

.063724 

.997068 

./6O 

.066655 

lo.CW 

.933345 

21 

40 

.064806 

18.  03 
17.98 

.997053 

.25 
.23 

.067752 

18^25 

.932248 

20 

41 
42 
43 

9.065885 
.066962 
.068036 

17.95 
17.90 

17  ftr* 

9.997039 
.997024 
.997009 

.25 
.25 

9.068846 
.069938 
.071027 

18.20 
18.15 

10.931154 

.930062 
.928973 

19 
18 
17 

44 
45 
46 
47 

.069107 
.070176 
.071242 
.072306 

1  1  .OO 

17.82 
17.77 
17.73 
17  fi7 

.996994 
.996979 
.996964 
.996949 

.25 
.25 
.25 
.25 

OK 

.072113 
.073197 
.074278 
.075356 

18.10 
18.07 
18.02 
17.97 

17  QQ 

.927887 
.926803 
.925722 
.924644 

16 
15 
14 
13 

48 

.073366 

l  <  .  o< 
17  63 

.996934 

.jew 

.076432 

i<  .yo 

.923568 

12 

49 

.074424 

1-r*  af) 

.996919 

*2? 

.077505 

17.88 

.922495 

11 

50 

.075480 

t  .OU 

17.55 

.996904 

,2o 
.25 

.078576 

17.85 
17.80 

.921424 

10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.076533 
.077583 
.078631 
.079676 
.080719 
.081759 
.082797 
.083832 
.084864 
9.085894 

17.50 
17.47 
17.42 
17.38 
17.33 
17.30 
17.25 
17.20 
17.17 

9.996889 
.996874 
.996858 
.996843 
.996828 
.996812 
.996797 
.996782 
.9967G6 
9.996751 

.25 

.27 
.25 
.27 
.27 
.25 

'.27 
.25 

9.079644 

.080710 
.081773 
.082833 
.083891 
.084947 
.086000 
.087050 
.088098 
9.089144 

17.77 
17.72 
17.67 
17.63 
17.60 
17.55 
17.50 
17.47 
17.43 

10.920356 
.919290 
.918227 
.917167 
.916109 
.915053 
.914000 
.912950 
.911902 
10.910856 

9 

8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  1".  !  Sine. 

D.  1". 

Cotang. 

D.1". 

Tang. 

' 

96° 


COSINES,   TANGENTS,   AND   COTANGENTS. 


' 

Sine. 

D.r. 

Cosine. 

D.  r. 

Tang. 

D.  1". 

Cotang. 

- 

0 

1 

2 

9.085894 
.086922 
.087947 

17.13 
17.08 

17  ftp* 

9.996751 
.996735 
.996720 

.27 
.25 

9.089144 
.090187 
.091228 

17.38 
17.35 

17  vj) 

10.910856 
.909813 
.908772 

60 
59 

58 

3 
4 
5 
6 
7 
8 
9 
10 

.088970 
.089990 
.091008 
.092024 
.093037 
.094047 
.095056 
.096062 

l  .UO 

17.00 
16.97  i 
16.93  , 
16.88 
16.83 
16.82 
16.77 
16.72 

.996704 
.996688 
.996673 
.996657 
.996641 
.996625 
.996610 
.996594 

'.27 
.25 
.27 
.27  i 
.27 
.25 
.27 
.27 

.092266 
.093302 
.094336 
.095367 
.096395 
.097422 
.098446 
.099468 

1<  .OU 

17.27 
17.23 
17.18 
17.13 
17.12 
17.07 
17.03 
16.98 

.907734 
.906698 
.905664 
.904633 
.903605 
.902578 
.901554 
.900532 

57 
56 
55 
54 
53 
52 
51 
50 

11 

9.097065 

1fi  PA 

9.996578 

97 

9.100487 

10.899513 

49 

12 
13 
14 
15 
16 

.098066 
.099065 
.100062 
.101056 
.102048 

10.  Oo 

16.65 
16.62 
16.57 
16.53 

.996562 
.996546 
.996530 
.996514 
.996498 

./Si 

.27 

.27 
.27 
.27 

.101504 
.102519 
.103532 
.104542 
.105550 

16!  92 
16.88 
16.83 
16.80 

.898496 
.897481 
.896468 
.895458 
.894450 

48 
47 
46 
45 
44 

17 
18 

.103037 
.104025 

16.48 
16.47 

.996482 
.996465 

.27 
.28 

.106556 
.107559 

16.77 
16.72 

.893444 
.892441 

43 
42 

19 
20 

.105010 
.105992 

16.42 
16.37 
16.35 

.996449 
.996433 

.27 
.27 
.27 

.108560 
.109559 

16.68 
16.65 
16.62 

.891440 
.890441 

41 
40 

21 

9.106973 

16  30 

9.996417 

9.110556 

16  58 

10.889444 

39 

22 
23 
24 

25 

26 
27 
28 

.107951 
.108927 
.109901 
.110873 
.111842 
.112809 
.113774 

16  '.27 
16.23 
16.20 
16.15 
16.12 
16.08 

.996400 
.996384 
.996368 
.996351 
.996335 
.996318 
.996302 

'.27 
.27 
.28 
.27 
.28 
.27 

.111551 
.112543 
.113533 
.114521 
.115507 
.116491 
.117472 

16^53 
16.50 
16.47 
16.43 
16.40 
16.35 

.888449 
.887457 
.886467 
.885479 
.884493 
.883509 
.882528 

38 
37 
36 
35 
84 
33 
32 

29 
30 

.114737 
.115698 

16^02 
15.97 

.996285 
.996269 

!27 

.28 

.118452 
.119429 

16.33 
16:28 
16.25 

.881548 
.880571 

31 

30 

31 

9.116656 

•IK  QK 

9.996252 

9ft 

9.120404 

1ft  99 

10.879596 

29 

32 
33 
34 
35 
36 
37 
38 
39 
40 

.117613 
.118567 
.119519 
.120469 
.121417 
.122362 
.123306 
.124248 
.125187 

10.  yo 
15.90 
15.87 
15.83 
15.80 
15.75 
15.73 
15.70 
15.65 
15.63 

.996235 
.996219 
.996202 
.996185 
.996168 
.996151 
.996134 
.996117 
.996100 

./SO 

.27 

.28 
.28 
.28 
.28 
.28 
.28 
.28 
.28 

.121377 
.122348 
.123317 
.124284 
.125249 
.126211 
.127172 
.128130 
.129087 

lo.SSe 

16.18 
16.15 
16.12 
16.08 
16.03 
16.02 
15.97 
15.95 
15.90 

.878623 
.877652 
.876683 
.875716 
.874751 
.873789 
.872828 
.87187'0 
.870913 

28 
27 
26 
25 
24 
23 
22 
21 
20 

41 

9.126125 

9.996083 

9Q 

9.130041 

1  P;  ftft 

10.869959 

19 

42 
43 
44 
45 
46 
47 

.127060 
.127993 
.128925 
.129854 
.130781 
.131706 

15.  5o 
15.55 
15.53 
15.48 
15.45 
15.42 

.996066 
.996049 
.996032 
.996015 
.995998 
.995980 

.Jso 
.28 
.28 
.28 
.28 
.30 

.130994 
.131944 
.132893 
.133839 
.134784 
.135726 

10.  oo 
15.83 
15.82 
15.77 
15.75 
15.70 

1  Pi  Rft 

.869006 
.868056 
.867107 
.866161 
.865216 
.864274 

18 
17 
16 
15 
14 
13 

48 

.132630 

15.40 

-j  e  oe 

.995963 

.28 

.136667 

10.  Oo 

15.63 

.863333 

12 

49 

.133551 

10.  OO 

.995946 

.28 

.137605 

.862395 

11 

50 

.134470 

15.32 
15.28 

.995928 

.30 

.28 

.138542 

15.62 
15.57 

.861458 

10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.  135387 
.136303 
.137216 
.138128 
.139037 
.139944 
.140850 
.141754 
.142655 
9.143555 

15.27 
15.22 
15.20 
15.15 
15.12 
15.10 
15.07 
15.02 
15.00 

9.995911 
.995894 
.995876 
.995859 
.995841 
.995823 
.995806 
.995788 
.995771 
9.995753 

.28 
.30 
.28 
.30 
.30 
.28 
.30 
.28 
.30 

9.139476 
.140409 
.141340 
.142269 
.143196 
.144121 
.145044 
.145966 
.  146885 
9.147803 

15.55 
15.52 
15.48 
15.45 
15.42 
15.38 
15.37 
15.32 
15.30 

10.860524 
.859591 
.858660 
.857731 
.856804 
.855879 
.854956 
.854034 
.853115 
10.852197 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

9 

Cosine. 

D.  1'. 

i  Sine.  • 

D.  1". 

Cotang. 

D.  1". 

Tang. 

' 

97»                                            82* 

TABLE   XII.      LOGARITHMIC    SItfES, 


' 

Sine. 

D.  1". 

Cosine. 

D.  1'. 

Tang. 

D.  1". 

Cotang. 

- 

0 

1 

2 

9.143555 
.144453 
.145349 

14.97 
14.93 

9.995753 
.995735 
.995717 

.30 
.30 

OA 

9.147803 
.148718 
.  149632 

15.25 
15.23 

10.852197 
.851282 
.850368 

60 
59 
58 

3 

.146243 

14.90 

.995699 

.OU 

.150544 

15.20 

.849456 

57 

4 

.  147136 

14.88 

14  SQ 

.995681 

'oc 

.151454 

15.17 

-jpr  -(  t 

.848546 

56 

5 
6 

7 
8 
9 

.148026 
.148915 
.149802 
.150686 
.  151569 

14.  oo 
14.82 
14.78 
14.73 
14.72 

.995664 
.995646 
.995628 
.995610 
.995591 

.*o 

.30 
.30 
.30 
.32 

Of) 

.152363 
.153269 
.154174 
.155077 
.155978 

10.  10 

15.10 
15.08 
15.05 
15.02 

.847637 
.846731 
.845826 
.844923 
.844022 

55 
54 
53 
52 
51 

10 

.152451 

14.70 
14.65 

.995573 

.OU 

.30 

.156877 

14^97 

.843123 

50 

11 

9.  153330 

9.995555 

OA 

9.157775 

14  Q3 

10.842225 

49 

12 
13 
14 
15 

.154208 
.155083 
.155957 
.156830 

14.  DO 

14.58 
14.57 
14.55 

.995537 
.995519 
.995501 
.995482 

.oU 
.30 
.30 
.32 

OA 

.158671 
.159565 
.160457 
.161347 

14  '.90 

14.87 
14.83 
14  82 

.841329 
.840435 
.839543 
.838653 

48 
47 
46 
45 

16 

.157700 

14.50 

.995464 

.OU 

QA 

.162236 

14  78 

.837764 

44 

17 

18 
19 
20 

.158569 
.159435 
.  160301 
.161164 

14.48 
14.43 
14.43 
14.38 
14.35 

.995446 
.995427 
.995409 
.995390 

.oU 

.32 
.30 
.32 
.30 

.163123 
.164008 
.164892 
.165774 

14.  to 

14.75 
14.73 
14.70 
14.67 

.836877 
.835992 
.835108 
.834226 

43 
42 

41 
40 

21 
22 

9.162025 

.162885 

14.33 

14  Qft, 

9.995372 
.995353 

.32 

qo 

9.166654 
.167532 

14.63 

14  fi9 

10.833346 
.832468 

39 

38 

23 
24 

.  163743 
.164600 

14.  OU 

14.28 

.995334 
.995316 

.  O/v 

.30 

.168409 
.169284 

14.  D/* 

14.58 

.831591 
.830716 

37 
36 

25 
26 

27 

.165454 
.166307 
.167159 

14.23 
14.22 
14.20 

.995297 
.995278 
.995260 

.32 
,32 
.30 

OO 

.170157 
.171029 
.171899 

14.55 
14.53 
14.50 

14  47 

.829843 
.828971 
.828101 

35 
34 
33 

28 
29 
30 

.168008 
.168856 
.169702 

14.15 
14.13 
14.10 
14.08 

.995241 
.995222 
.995203 

.0,4 

.32 
.32 
.32 

.  172767 
.173634 
.174499 

14.  4  1 

14.45 
14.42 
14.38 

.827233 
.826366 
.825501 

32 
31 
30 

31 

9.170547 

14  0°. 

9.995184 

32 

9.175362 

14  37 

10.824638 

29 

32 
33 
34 
35 
36 
37 
38 
39 

.171389 
.172230 
.173070 
.173908 
.174744 
.175578 
.176411 
.177242 

14.  UO 

14.02 
14.00 
13.97 
13.93 
13.90 
13.88 
13.85 

.995165 
.  995146 
.995127 
.995108 
.995089 
.995070 
.  995051 
.995032 

!32 
.32 
.32 
.32 
.32 
.32 
.32 

OO 

.176224 
.177084 
.177942 
.178799 
.179655 
.180508 
.181360 
.182211 

14^33 
14.30 
14.28 
14  27 
14.22 
14.20 
14.18 
Mi  °. 

.823776 
.822916 
.822058 
.821201 
.820345 
.819492 
.818640 
.817789 

28 
27 
26 
25 
24 
23 
22 
21 

40 

.178072 

13.83 
13.80 

.995013 

.O/* 

.33 

.183059 

.  lo 

14.13 

.816941 

20 

41 
42 
43 

9.178900 
.179726 
.180551 

13.77 
13.75 

9.994993 
.994974 
.994955 

.32 
.32 

qq 

9.183907 
.184752 
.185597 

14.08 
14.08 

10.816093 
.815248 
.814403 

19 
18 
17 

44 

.181374 

13.72 

1  °.  7rt 

.994935 

.00 

.186439 

14.03 

14  A9 

.813561 

16 

45 

46 

47 

.182196 
.183016 
.183834 

lo.  t(J 
13.67 
13.63 

.994916 
.9948% 
.994877 

!33 

.32 

qq 

.187280 
.188120 
.188958 

14.U/* 

14.00 
13.97 

.812720 
.811880 
.811042 

15 
14 
13 

48 
49 
50 

.184651 
.185466 
.186280 

13^58 
13.57 
13.53 

.994857 
.994838 
.994818 

.00 

.32 
.33 

.33 

.189794 
.190629 
.191462 

13^92 
13.88 
13.87 

.810206 
.809371 
.808538 

12 
11 
10 

51 

52 

9.187092 
.187903 

13.52 

9.994798 
.994779 

.32 

qq 

9.192294 
.193124 

13.83 

10.807706 

.806876 

9 

8 

53 
54 

.188712 
.189519 

13.48 
13.45 

.994759 
.994739 

.OO 

.33 

.193953 
.194780 

13.'78 

.806047 
.805220 

7 
6 

55 
56 

.190325 
.191130 

13.43 
13.42 

.994720 
.994700 

!33 

qq 

.195606 
.196430 

13.  77 
13.73 

•jq  r-f) 

.804394 
.803570 

5 
4 

57 

.191933 

13.38 

.994680 

.00 
OO 

.197253 

lo.  i  ~ 
•jq  ao 

.802747 

3 

58 
59 

.192734 
.193534 

13.35 
13.33 

.994660 
.994640 

.00 

.33 

.198074 
.198894 

lo.  Do 

13.67 

•jq  ae 

.801926 
.801106 

2 

1 

60 

9.194332 

13.30 

9.994620 

• 

9.199713 

lo.DO 

10.800287 

0 

' 

Cosine. 

D.r. 

Sine.  1  D.  1". 

Cotang.  ]  D.  1". 

Tang. 

' 

COSINES,    TANGENTS,    AN1>   COTANGENTS. 


/ 

Sine. 

D.  r. 

Cosine. 

D.  1'. 

Tang. 

D.  1". 

Cotang. 

/ 

0 

I 

2 
3 
4 
5 
6 
7 
8 
9 

9.194332 
.195129 
.195925 
.196719 
.197511 
.198302 
.  199091 
.199879 
.200666 
.201451 

13.28 
13.27 
13.23 
13.20 
13.18 
13.15 
13.13 
13.12 
13.08 

9.994620 
.994600 
.994580 
.994560 
.994540 
.994519 
.994499 
.994479 
.994459 
.994438 

.33 
.33 
.33 
.33 
.35 
.33 
.33 
.33 
.35 

9.199713 

.200529 
.201345 
.202159 
.202971 
.203782 
.204592 
.205400 
.206207 
.207013 

13.60 
13.60 
13.57 
13.53 
13.52 
13.50 
13.47 
13.45 
13.43 

10.800287 
.799471 
.798655 
.797841 
.797029 
.796218 
.795408 
.794600 
.793793 
.792987 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 

.202234 

13.05 
13.05 

.994418 

.33 
.33 

.207817 

13.40 
13.37 

.792183 

50 

11 

12 

9.203017 

.203797 

13.00 

-jo  (\(\ 

9.994398 
.994377 

.35 

OQ 

9.208619 
.209420 

13.35 

•jo  qq 

10.791381 
.790580 

49 

48 

18 

.204577 

lo.UU 
19  Qf^ 

.994357 

.OO 
QK 

.210220 

16.66 
•<  q  qrv 

.789780 

47 

14 

.205354 

i/«.yo 

19  Qf^ 

.994336 

.oO 

qq 

.211018 

lo.ou 

-jq  no 

.788982 

46 

15 
16 

17 
18 
19 
20 

.206131 
.206906 
.207679 
.208452 
.209222 
.209992 

I/*.VK> 

12.92 
12.88 
12.88 
12.83 
12.83 
12.80 

.994316 
.994295 
.994274 
.994254 
.994233 
.994212 

.OO 

.35 
.35 
.33 
.35 
.35 
.35 

.211815 
.212611 
.213405 
.214198 
.214989 
.215780 

lo.^o 
13.27 
13.23 
13.22 
13.18 
13.18 
13.13 

.788185 
.787389 
.786595 
.785802 
.785011 
.784220 

45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.210760 
.211526 
.212291 
.213055 
.213818 
.214579 
.215338 
.216097 
.216854 
.217609 

12.77 
12.75 
12.73 
12.72 
12.68 
12.65 
12.65 
12.62 
12.58 
12.57 

9.994191 
.994171 
.994150 
.994129 
.994108 
.994087 
.994066 
.994045 
.994024 
.994003 

.33 
.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 

9.216568 
.217356 
.218142 
.218926 
.219710 
.220492 
.221272 
.222052 
.222830 
.223607 

13.13 
13.10 
13.07 
13.07 
13!  03 
13.00 
13.00 
12.97 
12.95 
12.92 

10.783432 

.782644 
.781858 
.781074 
.780290 
.779508 
.778728 
.777948 
.777170 
.776393 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 

9.218363 
.219116 
.219868 
.220618 
.221367 
.222115 
.222861 

12.55 
12.53 
12.50 
12.48 
12.47 
12.43 

9.993982 
.993960 
.993939 
.993918 
.993897 
.993875 
.993854 

.37 
.35 
.35 
.35 
.37 
.35 

9.224382 
.225156 
.225929 
.226700 
.227471 
.228239 
.229007 

12.90 
12.88 
12.85 
12.85 
12.80 
12.80 

10.775618 
.774844 
.774071 
.773300 
.772529 
.771761 
.770993 

29 
28 
27 
26 

24 
23 

38 
39 

.223606 
.224349 

12.42 
12.38 

19  Qft 

.993832 
.993811 

.37 
.35 

07 

.229773 
.230539 

12.77 
12.77 

.770227 
.769461 

?? 

40 

.225092 

1/6.  Go 

12.35 

.993789 

.01 

.35 

.231302 

12.72 
12.72 

.768698 

20 

41 

42 

9.225833 
.226573 

12.33 

19  QA 

9.993768 
.993746 

.37 

OK 

9.232065 
.232826 

12.68 

19  fi7 

10.767935 
.767174 

19 

18 

43 
44 
45 
46 

47 
48 
49 
50 

.227311 
.228048 
.228784 
.229518 
.230252 
.230984 
.231715 
.232444 

l/«.  oU 

12.28 
12.27 
12.23 
12.23 
12.20 
12.18 
12.15 
12.13 

.993725 
.993703 
.993681 
.993660 
.993638 
.993616 
.993594 
.993572 

.GO 

.37 
.37 
.35 
.37 
.37 
.37 
.37 
.37 

.233586 
.234345 
.235103 
.235859 
.236614 
.237368 
.238120 
.238872 

l*.Oi 

12.65 
12.63 
12.60 
12.58 
12.57 
12.53 
12.53 
12.50 

.766414 
.765655 
.764897 
.764141 
.763888 
.762632 
.761880 
.761128 

17 
Iti 
15 
14 
13 
12 
11 
10 

51 

52 
53 
54 
55 
56 
57 
58 
59 
60 

9.233172 
.233899 
.234625 
.235349 
.236073 
.236795 
.237515 
.238235 
.238953 
9.239670 

12.12 
12.10 
12.07 
12.07 
12.03 
12.00 
12.00 
11.97 
11.95 

9.993550 
.993528 
.993506 
.993484 
.993462 
.993440 
.993418 
.993396 
.993374 
9.993351 

.37 
.37 
.37 
.37 
.37 
.37 
.37 
.37 
.38 

9.239622 
.240371 
.241118 
.241865 
.242610 
.843354 
.244097 
.544839 
.245579 
9.246319 

12.48 
12.45 
12.45 
12.42 
12.40 
12.38 
12.37 
12.33 
12.33 

10.760378 
.759629 
.758882 
.758135 
.757390 
.756646 
.755903 
.755161 
.754421 
10.753681 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

/ 

Cosine.   D.  1". 

Sine. 

D.  r. 

Cotang. 

D.I". 

Tang. 

/ 

89° 


10° 


TABLE   XII.      LOGARITHMIC    SINES, 


169* 


/ 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang. 

/ 

0 

1 

9.239670 
.240386 

11.93 

9.993351 
.993329 

.37 

9.246319 
.247057 

12.30 

10.753681  60 
..752943  59 

2 

.241101 

11  .92 

1  1  WR 

.993307 

.37 

QC 

04770.1    1Z.ZO 
.  <rt  t  1  w±    ^  o  or' 

.752206  58 

3 

4 
5 

.241814 
.242526 
.243237 

11  .  Oo 

11.87 
11.85 

noq 

.993284 
.993262 
.993240 

.00 

.37  ; 

.37  j 

qo   ! 

.  248530  H 
.249264 
.249998 

i¥».»i 

12.23 
12.23 

19  9ft 

.751470  57 
.750736  56 
.750002  :  55 

6 

.243947 

.00 

.993217 

.00 

.250730 

i/d  .4\J 

.749270  i  54 

7 

.244656 

11  .82 

.993195 

.37 

.251461 

12.18 

.748539  53 

8 

.245363 

11.78 

.993172 

.38 

.252191 

12.17 

.747809  52 

9 
10 

.246069 
.246775 

11.77 
11.77 
11.72 

.993149 
.993127 

.38 
.37 
.38 

.252920 
.253648 

12.15 
12.13 
12.10 

.747080  i  51 
.746352  j  5C 

11 

12 
13 
14 

9.247478 
:248181 
.248883 
.249583 

11.72 
11.70 
11.67 

9.993104 
.993081 
.993059 
.993036 

.38 
.37 

.38 

9.254374 
.255100 
.255824 
.256547 

12.10 
12.07 
12.05 

10.745626  i  49 
.744900  !  48 
.744176  1  47 
.743453  !  46 

15 

.250282 

11  .65 

.993013 

.38 

.257269 

12.03 

.742731  45 

16 
17 

.250980 
.251677 

11.63 
11.62 

.992990 
.992967 

.38 
.38 

.257990 
.258710 

12.02 
12.00 

.742010  44 
.741290  43 

18 

.252373 

11.60 

.992944 

.38 

.259429 

11  .98 

.740571  42 

19 
20 

.253067 
.253761 

11.57 
11.57 

.992921 
.992898 

.38 
.38 

.260146 
.260863 

11  .95 
11.95 

.739854 
.739137 

41 
40 

11.53 

.38 

11.92 

21 

9.254453 

9.99.2875 

9.261578 

10.738422 

39 

22 

.255144 

11.52 

.992852 

.38 

.262292 

11  .90 

.737708 

38 

23 
24 

•  .255834 
.256523 

11.50 
11.48 

.992829 
.992806 

.38 
.38 

.263005 
.263717 

11  .88 
11.87 

.736995 
.736283 

37 
36 

25 
26 

.257211 

.257898 

11.47 
11.45 

.992783 
.992759 

.38 
.40 

.264428 
.265138 

11.85 
11.83 

.735572 

.734862 

35 
34 

27 

.258583 

11.42 

.992736 

.38 

.265847 

11.82 

.734153 

33 

28 

.259268 

11.42 

.992713 

.38 

.266555 

11.80 

.733445 

32 

29 
30 

.259951 
.260633 

11.38 
11.37 

.992690 
.992666 

.38 
.40 

.267261 
.267967 

11.77 
11.77 

.732739 
.732033 

31 
30 

11.35 

.38 

11.73 

31 
32 

9.261314 
.261994 

11.33 

nqo 

9.992643 
.992619 

.40 

OQ 

9.268671 
.269375 

11.73 

n*7ft 

10.731329 
.730625 

29 

28 

33 
34 

.262673 
.263351 

.{HE 

11.30 

.992596 
.992572 

.€>O 

.40 

.270077 
.270779 

.  lU 

11.70 

.729923 
.729221 

27 
26 

35 

.264027 

11.27 

.992549 

.38 

.271479 

11.67 

.728521 

25 

36 

.264703 

11.27 

.992525 

.40 

.272178 

11.65 

.727822 

24 

37 

.265377 

11.23 

.992501 

.40 

.272876 

11.63 

.727124 

23 

38 
39 
40 

.266051 
.266723 
.267395 

11.23 
11.20 
11.20 
11.17 

.992478 
.992454 
.992430 

.38 
.40 
.40 
.40 

.273573 
.274269 
.274964 

11.62 
11.60 
11.58 
11.57 

.726427 
.725731 
.725036 

22 
21 

20 

41 
42 

9.268065 
.268734 

11.15 

9.992406 

.992382 

.40 

9.275658 
.276351 

11.55 

10.724342 

.723649 

19 

18 

43 
44 

45 
46 

47 
48 
49 
50 

.269402 
.270069 
.2707a5 
.271400 
.272064 
.272726 
.273388 
.274049 

11.13 
11.12 
11.10 
11.08 
11.07 
11.03 
11.03 
11.02 
10.98 

.992359 
.992335 
.992311 
.992287 
.992263 
.992239 
.992214 
.992190 

.38 
.40 
.40 
.40 
.40 
.40 
.42 
.40 
.40 

.277043 
.277734 

.278424 
.279113 
.279801 
.280488 
.281174 
.281858 

11.53 
11.52 
11.50 
11.48 
11.47 
11.45 
11.43 
11.40 
11.40 

.722957 
.722266 
.721576 
.7*20887 
.720199 
.719512 
.718826 
.718142 

17 
16 
15 
14 
13 
12 
11 
10 

51 

9.274708 

9.992166 

9.282542 

10.717458 

9 

52 
53 

.275367 
.276025 

10.98 
10.97 

.992142 
.992118 

.40 
.40 

.283225 
.283907 

11.38 
11.37 

.716775 
.716093 

8 

7 

54 

.276681 

10.93 

1ft  (M 

.992093 

.42 

AO   i 

.284588 

11.35 

nqq 

.715412 

6 

55 
56 
57 
58 
59 

.277337 
.277991 
.278645 
279297 
.279948 

lu.yo 
10.90 
10.90 
10.87 
10.85 

.992069 
.992044 
.992020 
.991996 
.991971 

,4U 

.42 

.40  [ 
.40 
.42 

.285268 
.285947 
.286624 
.287301 

.287977 

.00 

11.32 

11.28 
11.28 
11.27 

.714732 
.714053 
.713376 
.712699 
.712023 

5 
4 
3 
2 

1 

60 

9.280599 

10.85 

9.991947 

.40 

9.288652 

11  -25  10.711348 

0 

' 

Cosine. 

D.  1".  i!  Sine.  1  D..1". 

Cotang. 

D.  1'.    Tang.  |  ' 

100* 


COSINES,    TANGENTS,    AND    COTANGENTS. 


168* 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

9.280599 
.281248 
.281897 
.282544 
.283190 
.283836 
.284480 
.285124 
.285766 
.286408 
.287048 

10.82 
10.82 
10.78 
10.77 
10.77 
10.73 
10.73 
10.70 
10.70 
10.67 
10.67 

9.991947 
.991922 
.991897 
.991873 
.991848 
.991823 
.991799 
.991774 
.991749 
.991724 
.991699 

.42 
.42 
.40 
.42 
.42 
.40 
.42 
.42 
.42 
.42 
.42 

9.288652 
.289326 
.289999 
.290671 
.291342 
.292013 
.292682 
.293350 
.294017 
.294684 
.295349 

11.23 
11.22 
11.20 
11.18 
11.18 
11.15 
11.13 
11.12 
11.12 
11.08 

n.or 

10.711348 
.710674 
.710001 
.709329 
.708658 
.707987 
.707318 
.706650 
.705983 
.705316 
.704651 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.287688 
.288326 
.288964 
.289600 
.290236 
.290870 
.291504 
.292137 
.292768 
.293399 

10.63 
10.63 
10.60 
10.60 
10.57 
10.57 
10.55 
10.52 
10.52 
10.50 

9.991674 
.991649 
.991624 
.991599 
.991574 
.991549 
.991524 
.991498 
.991473 
.991448 

.42 
.42 
.42 
.42 
.42 
.42 
.43 
.42 
.42 
.43 

9.296013 
.296677 
.297339 
.298001 
.298662 
.299322 
.299980 
.300638 
.301295 
.301951 

11.07 
11.03 
11.03 
11.02 
11.00 
10.97 
10.97 
10.95 
10.93 
10.93 

10.703987 
.703323 
.702661 
.701999 
.701338 
.700678 
.700020 
.699362 
.698705 
.698049 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.294029 
.294658 
.295286 
.295913 
.296539 
.297164 
.297788 
.298412 
.299034 
.299655 

10.48 
10.47 
10.45 
10.43 
10.42 
10.40 
10.40 
10.37 
10.35 
10.35 

9.991422 
.991397 
.991372 
.991346 
'  .991321 
.991295 
.991270 
.991244 
.991218 
.991193 

.42 
.42 
.43 
.42 
.43 
.42 
.43 
.43 
.42 
.43 

9.302607 
.303261 
.303914 
.304567 
.305218 
.305869 
.306519 
.307168 
.307816 
.308463 

10.90 

10.88 
10.88 
10.85 
10.85 
10.83 
10.82 
10.80 
10.78 
10.77 

10.697393 
.696739 
.696086 
.695433 
.694782 
.694131 
.693481 
.692832 
.692184 
.691537 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 

32 
33 
34 
35 
36 
37 
38 
39 
40 

9.300276 
.300895 
.301514 
.302132 
.302748 
.303364 
.303979 
.304593 
.305207 
.305819 

10.32 
10.32 
10.30 
10.27 
10.27 
10.25 
10.23 
10.23 
10.20 
10.18 

9.991167 
.991141 
.991115 
.991090 
.991064 
.991038 
.991012 
.990986 
.990960 
.990934 

.43 
.43 
.42 
.43 
.43 
.43 
.43 
.43 
.48 
.43 

9.309109 
.309754 
.310399 
.311042 
.311685 
.312327 
.312968 
.313608 
.314247 
.314885 

10.75 
.  10.75 
10.72 
10.72 
10.70 
10.68 
10.67 
10.65 
10.63 
10.63 

10.690891 
.690246 
.689601 
.688958 
.688315 
.687673 
.687032 
.686392 
.685753 
.685115 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 

44 
45 
46 
47 

9.306430 
.307041 
.307650 
.308259 
.308867 
.309474 
.310080 

10.18 
10.15 
10.15 
10.13 
10.12 
10.10 

9.990908 
.990882 
.990855 
.990829 
.990803 
.990777 
.990750 

.43 
.45 
.43 
.43 
.43 
.45 

9.315523 
.316159 
.316795 
.317430 
.318064 
.318697 
319330 

10.60 
10.60 
10.58 
10.57 
10.55 
10.55 

10.684477 
.683841 
.683205 
.682570 
.681936 
.681303 
.680670 

19 
18 
17 
16 
15 
14 
13 

48 
49 
50 

.310685 
.311289 
.311893 

10.07 
10.07 
10.03 

.990724 
.990697 
.990671 

.45 
.43 
.43 

.319961 
.320592 
.321222 

10.52 
10.52 
10.50 
10.48 

.680039 
.679408 
.678778 

12 
11 
10 

51 

9.312495 

1fi  fiQ  ' 

9.990645 

At! 

9.321851 

in  /c? 

10.678149 

9 

52 
53 

.313097 
.313698 

10.02 

9  Oft 

.990618 
.990591 

.45 

.322479 
.323106 

10.45 

.677521 
.676894 

8 
7 

54 
55 
56 

.314297 
.314897 
.315495 

10.00 
9.97 

9QK 

.990565 
.990538 
.990511 

.45 
.45 

.323733 
.324358 
.324983 

10.42 
10.42 

.676267 
.675642 
.675017 

6 
5 
4 

57 
58 
59 
60 

.316092 
.316689 
.317284 
9.317879 

9.95 
9.92 
9.92 

.990485 
.990458 
.990431 
9.990404 

.45 
.45 
.45 

.325607 
.326231 
.326853 
9.327475 

10.40 
10.37 
10.37 

.674393 
.673769 
.673147 
10.672525 

3 
2 
1 

0 

' 

Cosine. 

D.I'. 

!  Sine. 

D.  1". 

Cotang.  D.  1'. 

Tang.   ' 

101- 


9HQ 


TABLE   XTT.       LOGAKTTHMTC    SINES, 


16T* 


' 

Sine. 

D.  1". 

Cosine. 

D.  r. 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 
3 
i 
5 

6 

7 
8 
9 

to 

9.317879 
.318473 
.319066 
.319658 
.320249 
.320840 
.321430 
.322019 
.322607 
.323194 
.323780 

9.90 

9.88 
9.87 
9.85 
9.85 
9.83 
9.82 
9.80 
9.78 
9.77 
9.77 

9.990404 
.990378 
.990351 
.990324 
.99029? 
.990270 
.990243 
.990215 
.990188 
.990161 
.990134 

.43 
.45 
.45 
.45 
.45 
.45 
.47 
.45 
.45 
.45 
.45 

9.327475 
.328095 
.328715 
.32;)334 
.329953 
.330570 
.331187 
.331803 
.332418 
.333033 
.333646 

10.33 
10.33 
10.32 
10.32 
10.28 
10.28 
10.27 
10.25 
10.25 
10.22 
10.22 

10.672525 
.671905 
.671285 
.670666 
.670047 
.669430 
.668813 
.668197 
.667582 
.666967 
.666354 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 

9.324366 

97°. 

9.990107 

9.334259 

10  20 

10.665741 

49 

12 
13 
14 
15 

16 

.324950 
.325534 
.326117 
.326700 
.327281 

.  <o 

9.73 
9.72 
9.72 
9.68 

9(\Q 

.990079 
.990052 
.990025 
.989997 
.989970 

.4f 
.45 
.45 
.47 
.45 

47 

.334871 
.335482 
.336093 
.336702 
.337311 

10^18 
10.18 
10.15 
10.15 

10  1  °. 

.665129 
.664518 
.663907 
.663298 
.662689 

48 
47 
46 
45 
44 

17 

18 
19 
20 

.327862 
.328442 
.329021 
.329599 

.Do 

9.67 
9.65 
9.63 
9.62 

.989942 
.989915 
.989887 
.989860 

i 

.45 

.47 
.45 

.47 

.337919 
.338527 
.339133 
.339739 

1U.  Id 

10.13 
10.10 
10.10 
10.08 

.662081 
.661473 

.660867 
.660261 

43 

41 

40 

21 
22 
23 
24 

9.330176 
.330753 
.831329 
.331903 

9.62 
9.60 
9.57 

9.989832 
.989804 
.989777 
.989749 

.47 
.45 
.47 

9.340344 
.340948 
.341552 
.342155 

10.07 
10.07 
10.05 

-\f\  no 

10.659656 
.659052 
.658448 
.657845 

£9 
36 

25 
26 

.332478 
.333051 

9.58 
9.55 

.989721 
.989693 

.47 
.47 

.342757 
.343358 

JU.Ud 

10.02 

.657243 
.656642 

35 
34 

27 
28 

.333624 
.334195 

9.55 
9.52 

9  CO 

.989665 
.989637 

.47 
.47 

AK 

.343958 
.344558 

10.00 
10.00 

90ft 

.656042 
.655442 

33 

b2 

29 

.334767 

.OO 

n  KQ 

.989610 

.40 

.345157 

.  JO 

9  97 

.654843 

31 

30 

.335337 

9!48 

.989582 

.47 
.48 

.345755 

9  '.97 

.654245 

oO 

31 

9.335906 

9.989553 

9.346353 

9QO 

10.653647 

29 

32 

.£36475 

947 

.989525 

.47 

.346949 

.  yd 

9  no 

.653051 

28 

33 

.337043 

.4< 

.989497 

Afl 

.347545 

.yo 

9Qq 

.652455 

27 

34 
35 

.337610 
.338176 

9.45 
9.43 

9451 

.989469 
.989441 

.47 
.47 

.348141 
.348735 

.yd 

9.90 

900 

.651859 
.651265 

26 
25 

36 

.338742 

.4d 

.989413 

JM 

.349329 

.  yu 

900 

.650671 

24 

37 

38 

.339307 
.339871 

9.42 
9.40 

9OQ 

.989385 
.989356 

.47 

.48 

.349922 
.£50514 

.00 

9.87 
907 

.650078 
.649486 

23 

22 

39 

.340434 

.OO 

.989328 

•  j' 

.351106 

.o< 

9  OK 

.648894 

21 

40 

.340996 

9.37 
9.37 

.989300 

.47 

.48 

.351697 

.  oO 

9.83 

.648303 

20 

41 
42 

9.341558 
.342119 

9.35 

9qq 

9.989271 
.989243 

.47 

4ft 

9.352287 
.352876 

9.82 

10.647713 
.647124 

19 
18 

43 

.342679 

.do 

.989214 

.4o 

.353465 

9ftrt 

.646535 

17 

44 
45 
46 

.343239 
.343797 
.344355 

9.33 
9.30 
9.30 

9  Oft 

.989186 
.989157 
.989128 

.47 

.48 

.48 

.354053 
.354640 
.355227 

.oU 

9.78 
9.78 

9r»v 

.645947 
.645360 
.644773 

16 
15 
14 

47 

.344912 

./6O 

9  oft 

.989100 

.47 

4ft 

.355813 

.  1  ( 
9»Jt 

.644187 

13 

48 
49 

.345469 
.346024 

.4& 

9.25 

.989071 
.989042 

.4o 

.48 

.356398 
.356982 

.  <O 

9.73 

.643602 
.643018 

12 
11 

50 

.346579 

9.25 
9.25 

.989014 

.47 
.48 

.357566 

9.73 
9.72 

.642434 

10 

51 

9.347134 

9.988985 

4ft 

9.358149 

9  70 

10.641851 

9 

52 

.3-47687 

9  .  33) 

.988956 

.4o 

.358731 

.641269 

8 

53 
54 
55 

.348240 
.348792 
.349343 

9.22 
9.20 
9.18 

91  7 

.988927 
.988898 
.988869 

.48 
.48 
.48 

4ft 

.359313 
.359893 
.360474 

9.70 
9.67 
9.68 

.640687 
.640107 
.639526 

7 
6 
5 

56 

.349893 

.  1  i 

.988840 

.4o 

.361053 

n'«~ 

.638947 

4 

57 
58 
59 

.350443 
.350992 
.351540 

9.17 
9.15 
9.13 
91°. 

.988811 
.988782 
.988753 

.48 
.48 
.48 

4ft 

.361632 
.362210 

.362787 

y  .  u5 

9.63 
9.62 

.638368 
.637790 
.637213 

3 
2 

1 

60 

9.352088 

.  lo 

9.988724 

.4o 

9.363364 

' 

10.636636 

0 

' 

Cosine. 

D.  1  .  1 

Sine, 

D.  1'. 

Co  tang. 

D.  r. 

Tang. 

' 

COSINES,    TANGENTS,    AND   COIM  NGENTS* 


' 

Sine. 

D.  1'. 

Cosine. 

D.  r. 

Tang. 

D.  1'. 

Cotang. 

' 

0 

1 

2 
3 
4 

5 

6 

7 
8 

9.352088 
.352635 
.353181 
.353726 
.354271 
.354815 
.355358 
.355901 
.356443 

9.12 
9.10 
9.08 
9.08 
9.07 
9.05 
9.05 
9.03 

9.988724 
.988695 
.988666 
.988636 
.988607 
.988578 
.988548 
.988519 
.988489 

.48 
.48 
.50 
.48 
.48 
.50 
.48 
.50 

4ft 

9.363364 
.363940 
.364515 
.365090 
.365664 
.366237 
.366810 
.367382 
.367953 

9.60 

9.58 
9.58 
9.57 
9.55 
9.55 
9.53 
9.52 

9KO 

10.636636 
.636060 
.635485 
.634910 
.634336 
.633763 
.633190 
.632618 
.632047 

60 
59 
58 
57 
56 
55 
54 
53 
52 

9 

10 

.356984 
.357524 

9^00 
9.00 

.988460 
.988430 

.4o 

.50 

.48 

.368524 
.369094 

,O/e 

9.50 
9.48 

.631476 
.630906 

51 
50 

11 
12 

9.358064 
.358603 

8.98 

8Q7 

9.988401 
.988371 

.50 

4ft 

9.369663 
.370232 

9.48 

9AK 

10.630337 
.629768 

49 

48 

13 
14 

.359141 
.359678 

.  ul 

8.95 

.988342 
.988312 

.40 

.50 

.370799 
.371367 

.40 

9.47 

.629201 
.628633 

47 

46 

15 

.360215 

8.95 

8QK 

.988282 

.50 

Kf) 

.371933 

9.43 

.628067 

45 

16 
17 
18 
19 

.360752 
.361287 
.361822 
.362356 

.yo 
8.92 
8.92 
8.90 

8OQ 

.988252 
.988223 
.988193 
.988163 

.ou 
.48 
.50 
.50 

tA 

.372499 
.373064 
.373629 
.374193 

9^42 
9.42 
9.40 

9OQ 

.627501 
.626936 
.626371 

.625807 

44 
43 
42 
41 

20 

.362889 

.OO 

8.88 

.988133 

.OU 

.50 

.374756 

.Ou 

9.38 

.625244 

40 

21 
22 
23 

9.363422 
.363954 
.364485 

8.87 
8.85 

9.988103 
.988073 
.988043 

.50 
.50 

tA 

9.375319 
.375881 
.376442 

9.37 
9.35 

9°e 

10.624681 
.624119 
.623558 

39 
38 
37 

24 

.365016 

8.85 

8OQ 

.988013 

.OU 

CA 

.377003 

.OO 

900 

.622997 

36 

25 
26 
27 
28 
29 
30 

.365546 
.366075 
.366604 
.367131 
.367659 
.368185 

.00 
8.82 
8.82 
8.78 
8.80 
8.77 
8.77 

.987983 
.987953 
.987922 
.987892 
.987862 
.987832 

.OU 

.50 
.52 
.50 
.50 
.50 
.52 

.377563 
.378122 
.378681 
.379239 
.379797 
.380354 

.00 

9.32 
9.32 
9.30 
9.30 
9.28 
9.27 

.622437 
.621878 
.621319 
.620761 
.620203 
.619646 

35 
34 
33 
82 
31 
SO 

31 
32 
33 
34 
35 
36 
37 

9.368711 
.369236 
.369761 
.370285 
.370808 
.371330 
.371852 

8.75 
8.75 
8.72 
8.72 
8.70 
8.70 

9.987801 
.987771 
.987740 
.987710 
.987679 
.987649 
.987618 

.50 
.52 
.50 
.52 
.50 
.52 

tA 

9.380910 
.381466 
.382020 
.382575 
.383129 
.383682 
.384234 

9.27 
9.23 
9.25 
9.23 
9.22 
9.20 

10.619090 
.618534 
.617980 
.617425 
.616871 
.616318 
.615766 

29 
28 
27 
26 
25 
24 
23 

38 
39 

.372373 
.372894 

8^68 

8A7 

.987588 
.987557 

.ou 
.52 

.384786 
.385337 

&*18 

91ft 

.615214 
.614663 

22 
21 

40 

.373414 

.Ol 

8.65 

.987526 

!so 

.385888 

.  lo 

9.17 

.614112 

?0 

41 
42 
43 

44 
45 
46 

9.373933 
.374452 
.374970 
.375487 
.376003 
.376519 

8.65 
8.63 
8.62 
8.60 
8.60 

9.987496 
.987465 
.987434 
.987403 
.987372 
.987341 

.52 
.52 
.52 
.52 

.52 

9.386438 
.386987 
.387536 
.388084 
.388631 
.389178 

9.15 
9.15 
9.13 
9.12 
9.12 

10.613562 
.613013 
.612464 
.611916 
.611369 
.610822 

19 

18 
17 
16 
15 
14 

47 
48 
49 
50 

.377035 
.377549 
.378063 
.378577 

8.57 
8.57 
8.57 
8.53 

.987310 
.987279 
.987248 
.987217 

.52 
.52 
.52 

.52 
.52 

.389724 
.390270 
.390815 
.391360 

9.10 
9.10 
9.08 
9.08 
9.05 

.610276 
.609730 
.609185 
.608640 

13 
12 
11 
;  10 

51 
52 
53 
54 
55 
56 
57 

9-379089 
.379601 
.380113 
.380624 
.381134 
.381643 
.382152 

8.53 
8.53 
8.52 
8.50 

8.48 
8.48 

8:10 

9.987186 
.987155 
.987124 
.987092 
.987061 
.987030 
.986998 

.52 
.52 
.53 
.52 
.52 
.53 

9.391903 
.392447 
.392989 
.393531 
.394073 
.394614 
.395154 

9.07 
9.03 
9.03 
9.03 
9.02 
9.00 

9AA 

10.608097 
.607553 
.607011 
.606469 
.605927 
.605386 
.604846 

9 

8 
7 
i  6 
5 
4 
3 

58 

"  .382661 

.4o 

.986967 

•  j~ 

.395694 

.UU 

.604306 

2 

59 

.383168 

8.45 

.986936 

,5% 

.396233 

,8.98 

.603767   1 

60 

9.383675 

8.45 

9.986904 

.53 

|  9.396771 

8.97 

10.603229   0 

' 

Cosine. 

D.  r. 

Sine. 

D.  1".  ||  Cotang. 

D.  1".    Tang. 

' 

103° 


211 


TABLE   XII.       LOGARITHMIC    SIKES, 


165° 


' 

Sine. 

D.I'. 

Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

2 
3 

4 

9.383675 

.384182 
.384687 
.385192 
.385697 

8.45 
8.42 
8.42 

8.42 

9.986904 
.986873 
.986841 
.986809 
.986778 

.52 
.53 
.53 
.52 

KO 

9.396771 
.397309 
.397846 
.398383 
.398919 

8.97 
8.95 
8.95 
8.93 

8Q9. 

10.603229 
.602691 
.602154 
.601617 
.601081 

60 
59 

58 
57 
56 

5 

.386201 

8  op 

.986746 

.Do 
to 

.399455 

.yo 

.600545 

55 

6 

.386704 

.OO 

800 

.986714 

.Oo 

KO 

.399990 

8QA 

.600010 

54 

7 
8 

.387207 
.387709 

.OO 

8.37 

8  OK 

.986683 
.986651 

.(XV 

.53 

.400524 
.401058 

.  yu 
8.90 

800 

.599476 
.598942 

53 

52 

9 

10 

.388210 
.388711 

.oO 

8.35 
8.33 

.986619 
.986587 

!53 
.53 

.401591 
.402124 

.00 

8.88 
8.87 

.598409 
.597876 

51 
50 

11 
12 

9.389211 
.389711 

8.33 

89.9 

9.986555 
.986523 

.53 

to 

9.402656 
.403187 

8.85 

8  OK 

10.597344 
.596813 

49 

48 

13 

.390210 

.66 

80ft 

.986491 

.Oo 

KO 

.403718 

.00 

8  OK 

.596282 

47 

14 

.390708 

.oU 

.986459 

.Oo 

KO 

.404249 

.oO 

8Q9 

•.595751 

46 

15 

16 

.391206 
.391703 

8.30 
8.28 

897 

.986427 
.986395 

.Do 

.53 

to 

.404778 
.405308 

.CVi 

8.83 

8  on 

.595222 
.594692 

45 

44 

17 

.392199 

.«4 

897 

.986363 

.Do 

KQ 

.405836 

.OU 
ft.  P.O 

.594164 

43 

18 
19 

.392695 
.393191 

.*» 

8.27 

.986331 
.986299 

.Oo 

.53 

KK 

.406364 
.406892 

o.OU 

8.80 

.593636 
.593108 

42 
41 

20 

.393685 

8^23 

.986266 

.DO 

.53 

.407419 

8.77 

.592581 

40 

21 
22 
23 
24 
25 

9.394179 
.394673 
.395166 
.395658 
.396150 

8.23 

8.22 
8.20 
8.20 

9.986234 
.986202 
.986169 
.986137 
.986104 

.53 
.55 
.53 
.55 

KO 

9.407945 
.408471 
.408996 
.409521 
.410045 

8.77 
8.75 
8.75 
8.73 

87°. 

10.592055 
.591529 
.591004 
.590479 
.589955 

39 
38 
37 
36 
35 

26 

27 

.396641 
.397132 

8.18 
8.18 

8-JK 

.986072 
.986039 

.Do 

.55 

to 

.410569 
.411092 

.  <o 

8.72 

879 

.589431 

.588908 

34 
33 

28 

.397621 

.  10 
Q  -tn 

.986007 

.Do 

.411615 

.  <  «i 

87ft 

.588385 

32 

29 
30 

.398111 
.398600 

8.'  15 
8.13 

.985974 
.985942 

.55 

.53 
.55 

.412137 
.412658 

.  <U 

8.68 
8.68 

.587863 
.587342 

31 
30 

31 

9.399088 

819 

9.985909 

KK 

9.413179 

o  ai-r 

10.586821  29 

32 
33 
34 
35 
36 

.399575 
.400062 
.400549 
.401035 
.401520 

.I* 

8.12 
8.12 
8.10 
8.08 

8  no 

.985876 
.985843 
.985811 
.985778 
.985745 

.00 

.55 
.53 
.55 

.55 
5- 

.413699 
.414219 
.414738 
.415257 
.415775 

o.o7 
8.67 
8.65 
8.65 
8.63 
8R°. 

.586301 
.585781 
.585262 
.584743 
.584225 

28 
27 
26 
25 
24 

37 

.402005 

.Uo 

.985712 

) 

KK 

.416293 

.Do 

.583707 

23 

38 
39 
40 

.402489 
.402972 
.403455 

8io5 
8.05 
8.05 

.985679 
.985646 
.985613 

.DO 

.55 
.55 
.55 

.416810 
.417326 

.417842 

s!eo 

8.60 
8.60 

.583190 
.582674 
.582158 

22 
21 
20 

41 
42 

9.403938 
.404420 

8.03 

9.985580 
.985547 

.55 

9.418358 

.418873 

8.58 

10.581642 
.581127 

19 

18 

43 

.404901 

8.02 

8ft9 

.985514 

.55 

.419387 

8.57 

8K7 

.580613 

17 

44 

45 

.405382 
.405862 

.UJ* 

8.00 

.985480 
.985447 

!55 

.419901 
.420415 

.Of 

8.57 

8KK 

.580099 
.579585 

16 
15 

46 

.406341 

r  '  Q 

.985414 

.55 

KK 

.420927 

.DO 

O  tt 

.579073 

14 

47 
48 

.406820 
.407299 

7.  '98 

7  Q7 

.985381 
.985347 

.DO 

.57 

KK 

.421440 
.421952 

O.OD 

8.53 

,578560 

.578048 

13 
12 

49 

.407777 

*  .y< 

7QK 

.985314 

.DO 

K.7 

.422463 

CKf 

.577'537 

11 

50 

.408254 

.  yo 
7.95 

.985280 

.Of 

.55 

.422974 

8^50 

.577026 

10 

51 

9.408731 

9.985247 

K7 

9.423484 

10.576516 

9 

52 
53 
54 
55 
56 

.409207 
.409682 
.410157 
.410632 
.411106 

7^93 

7.92 
7.92 
7.90 

.985213 
.985180 
.985146 
.985113 
.985079 

.Of 

.55 
.57 
.55 

.57 

.423993 
.424503 
.425011 
.425519 

.426027 

8.48 
8.50 
8.47 
8.47 
8.47 

84K 

.576007 
.575497 
.574989 
.574481 
.573973 

8 
7 
6 
5 
4 

57 

.411579 

7.88 

7  ftp. 

.985045 

•j** 

.426534 

.40 

8AK 

.573466 

3 

58 
59 
60 

.412052 
.412524 
9.412996 

(  .00 

7.87 
7.87 

.985011 
.984978 
9.9S4944 

!55 
.57 

.427041 
.427547 
9.428052 

.40 

8.43 

8.42 

.572959 
.572453 
10.571948 

2 

1 
0 

1  \  Cosine. 

D.  r.  1 

Sine. 

D.  1'.  II  Cotang.  D.  1'. 

Tang. 

i 

15° 


COSINES,    TANGENTS,    AND    COTANGENTS. 


164* 


/ 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

/ 

0 

1 

2 

9.412996 
.413467 
.413938 

7.85 
7.85 

9.984944 
.984910 
.984876 

.57 
.57 

9.428052 

.428558 
.429062 

8.43 
8.40 

10.571948 
.571442 
,570938 

60 
59 

58 

3 

.414408 

7.83 

.984842 

.57 

.429566 

Q-^rt 

.570434 

57 

4 

.414878 

7.83 

.984808 

tr» 

.430070 

8  OQ 

.569930 

56 

5 

6 

.415347 
.415815 

7.82 
7.80 

.984774 
.984740 

.57 
.57 

.430573 
.431075 

.OO 

8.37 
807 

.569427 
.568925 

55 
54 

7 

.416283 

7.80 

.984706 

.57 

.431577 

.04 
807 

.568423 

53 

8 
9 
10 

.416751 
.417217 
.417684 

7.80 
7.77 
7.78 

7.77 

.984672 
.984638 
.984603 

.57 
.57 
.58 
.57 

.432079 
.432580 
.433080 

.Of 

8.35 
8.33 
8.33 

.567921 
.567420 
.566920 

52 
51 
50 

11 

9.418150 

9.984569 

9.433580 

800 

10.566420 

49 

12 

.418615 

7.75 

.984535 

.57 

to 

.434080 

.00 
809 

.565920 

48 

13 

.419079 

7.73 

.984500 

.Do 

.434579 

.o<« 
809 

.565421 

47 

14 

.419544 

7.75 

.984466 

.57 

.435078 

.04 

.564922 

46 

15 

.420007 

7.72 

.984432 

.57 

to 

.435576 

8.30 
890 

.564424 

45 

16 

.420470 

7.72 

.984397 

.00 

.436073 

.<*o 

.563927 

44 

17 

.420933 

7.72 

.984363 

'  fft 

.436570 

8  "oft 

.563430 

43 

18 

.421395 

7.70 

.984328 

.00 

.437067 

./CO 

897 

.562933 

42 

19 
20 

.421857 
.422318 

7.70 
7.68 
7.67 

.984294 
.984259 

!58 
.58 

.437563 
.438059 

.4,1 

8.27 
8.25 

.562437 
,561941 

41 

40 

21 
22 
23 

9.422778 
.423238 
.423697 

7.67 
7.65 

9.984224 
.984190 
.984155 

.57 

.58 

to 

9.438554 
.439048 
.439543 

8.23 
8.25 
890 

10.561446 
.560952 
.560457 

39 
38 
37 

24 

.424156 

7.65 

.984120 

.00 

.440036 

.«* 

.559964 

36 

25 

.424615 

7.65 

.984085 

.58 

to 

.440529 

8.22 

899 

.559471 

35 

26 

27 

.425073 
.425530 

7.63 

7.62 

.984050 
.984015 

.00 
.58 

.441022 
.441514 

.66 

8.20 

.558978 
.558486 

34 
33 

28 

425987 

7.62 

.983981 

.57 

.442006 

0^0 

.557994 

32 

29 

.426443 

7.60 

.983946 

.58 

.442497 

8.18 

.557503 

31 

30 

.426899 

7.60 
7.58 

.983911 

.58 
.60 

.442988 

8.18 
8.18 

.557012 

30 

31 

9.427354 

9.983875 

9.443479 

10.556521 

29 

32 
33 
34 
35 

.427809 
.428263 
.428717 
.429170 

7.58 
7.57 
7.57 
7.55 

.983840 
.983805 
.983770 
.983735 

.58 
.58 
.58 
.58 

.443968 
.444458 
.444947 
.445435 

8.15 
8.17 
8.15 
8.13 

.556032 
.555542 
.555053 
.554565 

28 
27 
26 
25 

36 
37 
38 
39 
40 

.429623 
.430075 
.430527 
.430978 
.431429 

7.55 
7.53 
7.53 

7.52 
7.52 
7.50 

.983700 
.983664 
.983629 
.983594 
.983558 

.58 
.60 
.58 
.58 
.60 
.58 

.445923 
.446411 
.446898 
.447384 

.447870 

8.13 
8.13 
8.12 
8.10 
8.10 
8.10 

.554077 
.553589 
.553102 
.552616 
.552130 

24 
23 
22 
21 
20 

41 

9.431879 

9.983523 

Art 

9.448356 

8rtQ 

10.551644 

19 

42 
43 
44 
45 

.432329 
.432778 
.4a3226 
.433675 

7.50 

7.48 
7.47 

7.48 

.983487 
.983452 
.983416 
.983381 

.OU 

.58 
.60 

.58 

.448841 
.449326 
.449810 
.450294 

.Uo 

8.08 
8.07 
8.07 

.551159 
.550674 
.550190 
.549706 

18 
17 
16 
15 

46 
47 
48 
49 
50 

.434122 
.434569 
.435016 
.435462 
.435908 

7.45 
7.45 
7.45 
7.43 
7.43 
7.42 

.983345 
.983309 
.983273 
.983238 
.983202 

.60 
.60 
.60 
.58 
.60 
.60 

.450777 
.451260 
.451743 
.452225 
.452706 

8.05 
8.05 
8.05 
8.03 
8.02 
8.02 

.549223 
.548740 
.548257 
.547775 
.547294 

14 

13 
12 
11 
10 

51 
52 

9.436353 
.436798 

7.42 

9.983166 
.983130 

.60 

1  9.453187 
.453668 

8.02 

10.546813 
.546332 

9 

8 

53 

54 

.437242 

.437685 

7.40 
7.40 

.983094 
.983058 

.60 
.60 

1  .454148 

1  .454628 

8.00 
8.00 

.545852 
.545372 

7 
6 

55 
56 

.438129 
.438572 

7.38 
7.38 

.983022 
.982986 

.60 
.60 

.455107 
.455586 

7.  "98 

.544893 
.544414 

5 
4 

57 
58 

.439014 
.439456 

7.37 
7.37 

.982950 
.982914 

.60 
.60 

.456064 
.456542 

7.97 
7.97 

.543936 
.543458 

3 
2 

59 
60 

.439897 
9.440338 

7.35 
7.35 

.982878 
9.982842 

.60 
.60 

.457019 
9.457496 

7.95 
7.95 

.542981 
10.542504 

1 
0 

'  |  Cosine. 

D.  1". 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

1 

105" 


16* 


TABLE   XII.      LOGARITHMIC   SINES, 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 

9.440338 

.440778 
.441218 

7.33 
7.33 
7  33 

9.982842 
.982805 
.982769 

.62 
.60 
fin 

9.457496 
.457973 
.458449 

7.95 
7.93 

10.542504 
.542027 
.541551 

60 
59 
58 

3 

.441658 

7  30 

.982733 

.ou 

.458925 

7.93 

.541075 

57 

4 
5 
6 

.442096 
.442535 
.442973 

7.  '32 
7.30 

7  28 

.982696 
.982660 
.982624 

!eo 

.60 

.459400 
.459875 
.460349 

7.92 
7.92 
7.90 

.540600  56 
.540125  55 
.539651  54 

7 
8 
9 
10 

.443410 
.443847 
.444284 
.444720 

7.'28 
7.28 
7.27 
7.25 

.982587 
.982551 
.982514 
.982477 

!eo 

.62 
.62 
.60 

.460823 
.461297 
.461770 
.462242 

7.90 
7.90 

7.88 
7.87 
7.88 

.539177 
.538703 
.538230 
.537758 

53 
52 
51 
50 

11 

9.445155 

7.25 

9.982441 

oa 

9.462715 

7  OK 

10.537285 

49 

12 
13 

.445590 
.446025 

7.  '25 

7  23 

.982404 
.982367 

.0/4 

.62 

AO 

.463186 
.463658 

.ou 

7.87 

7  ftp 

.536814  48 
.536342  !  47 

14 

.446459 

7'  oq 

.982331 

.ou 

.464128 

1  .00 

.535872  46 

15 

.446893 

i  ./4o 

7  22 

.982294 

P.9 

.464599 

7.85 

.535401  45 

16 
17 

.447326 
.447759 

7^22 
7  20 

.982257 
.982220 

,O/4 

.62 

on 

.465069 
.465539 

7^83 

7  ft9 

.534931  44 
.534461  i  43 

18 

.448191 

7  20 

.982183 

.0.4 

.466008 

1  .04 

.533992  42 

19 
20 

.448623 
.449054 

7.'  18 
7.18 

.982146 
.982109 

!62 
.62 

.466477 
.466945 

7  '.80 
7.80 

.533523 
.533055 

41 
40 

21 
22 
23 
24 
25 
26 

9.449485 
.449915 
.450345 
.450775 
.451204 
.451632 

7.17 
7.17 
7.17 
7.15 
7.13 

7  -jo 

9.982072 
.982035 
.981998 
.981961 
.981924 
.981886 

.62 
.62 
.62 
.62 
.63 

!  9.467413 
.467880 
1  .468347 
.468814 
.469280 
.469746 

7.78 
7.78 
7.78 

7.77 
7.77 

10.532587 
.532120 
.531653 
.531186 
.530720 
.530254 

39 

38 
37 
36 
35 
34 

27 

28 
29 

.452060 
.452488 
.452915 

1  .  lo 

7.13 

7.12 

719 

.981849 
.981812 
.981774 

!62 
.63 

.470211 
.470676 
.471141 

7.75 
7.75 
7.75 

7rvo 

.529789 
.529324 

.528859 

33 
32 
31 

30 

.453342 

<  .  1/4 

7.10 

.981737 

.62 
.62 

.471605 

.  10 

7.73 

.528395 

30 

31 

9.453768 

7  10 

9.981700 

9.472069 

10.527931 

29 

32 

.454194 

l  .  IU 

7  08 

.981662 

(\9 

.472532 

7  79 

.527468 

28 

33 

.454619 

7  no 

.981625 

.0/4 

.472995 

<  .  1/4 

.527005 

27 

34 

.455044 

.Uo 

7  Oft 

.981587 

.63 

an 

.473457 

7.70 

7  r-n 

.526543 

26 

35 

.455469 

<  .  Uo 
7  07 

.981549 

.Oo 

.473919 

7  7'0 

.526081 

25 

36 

.455893 

1  .Ul 

7  OK 

.981512 

co 

.474381 

l~  CO 

.525619 

24 

37 

.456316 

.UO 

7  05 

.981474 

.00 

.474842 

1  .OO 

f  00 

.525158 

23 

38 
39 

.456739 
.457162 

7.'05 

7  03 

.981436 
.981399 

!62 

/•o 

.475303 
.475763 

i  .OO 

7.67 

r-  art 

.524697 
.524237 

22 
21 

40 

.457584 

7!  03 

.981361 

.Oo 
.63 

.476223 

i  .VI 

7.67 

.523777 

20 

41 

9.458006 

7  09 

9.981323 

an 

9.476683 

7  ft*. 

10.523317 

19 

42 

.458427 

<  .  U/4 

.981285 

.Oo 

.477142 

1  .00 

.522858 

18 

43 
44 
45 

.458848 
.459268 
.459688 

7.'00 
7.00 

7  OO 

.981247 
.981209 
.981171 

.63 
.63 
.63 

an 

.477601 
.478059 
.47'8517 

7^63 
7.63 

7  P.P 

.522399 
.521941 
.521483 

17 
16 
15 

46 

.460108 

1  .UU 

6QQ 

.981133 

.Oo 

an 

.478975 

<  .Oo 
7  co 

.521025 

14 

47 

.460527 

.  yo 
6  98 

.981095 

.Oo 
CO 

.479432 

.0,4 

.520568 

13 

48 
49 

.460946 
.461364 

6^97 

.981057 
.981019 

.OO 

.63 

.479889 
.480345 

7^60 

.520111 
.519655 

12 

11 

50 

.461782 

6.97 
6.95 

.980981 

.63 

.65 

.480801 

7.60 
7.60 

.519199 

10 

51 

9.462199 

6  OK 

9.980942 

9.481257 

7KQ 

10.518743 

9 

52 

.462616 

.yo 

.980904 

OO 

.481712 

.Oo 

7Kft 

.518288 

8 

53 

.463032 

ft  OQ 

.980866 

.Oo 
CK 

.482167 

.Oo 

7  57 

.517833 

7 

54 

.463448 

a  'r\n 

.980827 

.OO 
an 

.482621 

.517379 

6 

55 

.463864 

o.yo 

.980789 

.Oo 

OK 

.483075 

7  try 

.516925 

5 

56 

.464279 

!?'r!^ 

.980750 

.00 

.483529 

.o< 

.516471 

4 

57 

.464694 

b.92 

.980712 

.63 

OK 

.483982 

7.55 

.516018 

3 

58 
59 

.465108 
.465522 

6.90 
6.90 

6QQ 

.980673 
.980635 

.OO 

.63 

.484435 

.484887 

7^53 

7  CO 

.515565 
.515113 

2 
1 

60 

9.465935 

.OO 

9.980596 

.65 

9.485339 

.Oo 

10.514661 

0 

' 

Cosine. 

D.I".  | 

Sine. 

D.  i".  i 

Cotang. 

D.  r.    Tang.    ' 

214 


73- 


COSINES,    TANGENTS,    AND   COTANGENTS. 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

9.4659a5 

9.980596 

9.485339 

7  Kq 

10.514661 

60 

1 
g 

3 
4 
5 

.466348 
.466761 
.467173 
.467585 
.467996 

6.88 
6.88 
6.87 
6.87 
6.85 

.980558 
.980519 
.980480 
.980442 
.980403 

!es 

.65 
.63 
.65 

.485791 
.486242 
.486693 
.487143 
.487593 

i  .Oo 

7.52 

7.52 
7.50 
7.50 

n  en 

.514209 
.513758 
.513307 
.512857 
.512407 

59 
58 
57 
56 
55 

6 

7 

.48S407 
.468817 

6.85 
6.83 

.980364 
.980325 

!es  \ 

.488043 
.488492 

<  .OIF 

7.48 
7  48 

.511957 
.511508 

54 
53 

8 

.469227 

6.83 

683 

.980286 

(•K 

.488941 

7  48 

.511059 

52 

9 

.469637 

.OO 

.980247 

.DO 

.489390 

7  47 

.510(510 

51 

10 

.470046 

6.82 
6.82 

.980208 

.'65 

.489838 

i  .**< 

7.47 

.510162 

50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.470455 
.470863 
.471271 
.471679 
.472086 
.472492 
.472898 
.473304 
.473710 
.474115 

6.80 
6.80 
6.80 
6.78 
6.77 
6.77 
6.77 
6.77 
6.75 
6.73 

9.980169 
.980130 
.980001 
.980052 
.980012 
.979973 
.979934 
.979895 
.979855 
.979816 

.65 
.65 
.65 
.67 
.65 
.65 
.65 
.67 
.65 
.67 

9.490286 
.4;)0733 
.491180 
.491627 
.492073 
.492519 
.492965 
.493410 
.493854 
.494299 

7.45 
7.45 
7.45 
7.43 
7.43 
7.43 
7.42 
7.40 
7.42 
7.40 

10.509714 
.509267 
.508820 
.508373 
.507927 
.507481 
.507035 
.506590 
.506146 
.505701 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 

22 

9.474519 
.474923 

6.73 

9.979776 
.979737 

•65 

9.494743 

.495186 

7.38 

10.505257 

.504814 

39 

38 

23 
24 

.475327 
.475730 

6.73 
6.72 

.979697 
.979658 

!65 

.495630 
.496073 

7*.88 

n  q7 

.504370 
.503927 

37 
36 

25 
26 

.476133 
.476536 

6.72 
6.72 

.979618 
.979579 

•65 

.496515 
.496957 

r  .01 

7.37 

.503485 
.503043 

35 
34 

27 

.476938 

6.70 

.979539 

.67 

.497399 

7.3< 

7  37 

.502601 

33 

28 

.477340 

6.70 

.979499 

.67 

.4D7841 

i  .01 

.502159 

32 

29 
30 

.477741 

.478142 

6.68 
6.68 
6.67 

.979459 
.979420 

.67 
.65 
.67 

.498282 
.498722 

7*.«8 

7.35 

.501718 
.501278 

31 
30 

31 

9.478542 

6   Off 

9.979380 

9.499163 

7  3.3. 

10.500837 

29 

32 

.478942 

.0< 

.979340 

•.5* 

.499603 

<  .00 

7  39 

.500397 

28 

33 

.479342 

6.67 

.979300 

.0< 

.500042 

1  .  O.4 

.499958 

27 

34 

.479741 

6.65 

.979260 

.67 

.500481 

7  32 

.499519 

26 

35 
36 

.480140 
.480539 

6.65 
6.65 

.979220 
.979180 

.67 
.67  • 

.500920 
.501359 

7^32 
7  SO 

.490080 
.498641 

25 
24 

37 

.480937 

6.63 

.979140 

.67 

.501797 

7  3A 

.498203 

23 

38 
39 

40 

.481334 

.481731 

.482128 

6.62 
6.62 
6.62 
6.62 

.979100 
.979059 
.979019 

!68 
.67 
.67 

.502235 

.502672 
.503109 

i  .oU 

7.28 
7.28 
7.28 

.497765 
.497328 
.49689J 

22 
21 
20 

41 

9.482525 

9.978979 

Off 

9.503546 

797 

10.496454 

19 

42 

.482921 

6.60 

.978989 

.0< 

.503982 

,Ai 

7  97 

.496018 

18 

43 

41 
45 

.483316 
.483712 

.484107 

6.58 
6.60 
6.58 

.978898 
.978858 
.978817 

.68 
.67 
.68 

.504418 
.504854 
.505289 

I  .&( 

7.27 

Z-25 

.495582 
.495146 
.494711 

17 
16 
15 

46 

.484501 

6.57 

.978777 

n£ 

.505724 

7  9K 

.49427'6 

14 

47 
48 

.484895 
.485289 

6.57 
6.57 

.978737 
.978696 

.0< 
.08 

OO 

.506159 
.506593 

7  '.23 

.493841 
.493407 

13 
12 

49 

.485682 

6.55 

.978655 

.Oo 

.507027 

r*v2 

.492973 

11 

50 

.486075 

6.55 
6.53 

.978615 

.67 
.68 

.507460 

^  .M 
7.22 

.492540 

10 

51 

9.486467 

9.978574 

9.507893 

r»  oo 

10.492107 

9 

52 

.486860 

6.55 

.978533 

'«» 

.508326 

7  99 

.491674 

8 

53 
54 
55 
56 

.487251 
.487643 
.488034 

.488424 

6.52 
6.53 
6.52 
6.50 

.978493 
.978452 
.978411 
.978370 

.07 
.68 
.68 
.68 

.508759 
.509191 
.509622 
.510054 

7^20 
7.18 
7.20 

7  1  Q 

.491241 
.490809 
.490378 
.489946 

7 
6 
5 
4 

57 

.488814 

6.50 

978329 

*«2 

.510485 

<  .  lo 
7  1  Q 

.489515 

3 

58 

.489204 

6.50 

.978288 

.08 

AQ 

.510916 

i  .lo 
717 

.489084 

2 

59 

.489593 

6.48 

.978247 

.Do 

.511346 

1  .11 

.488654 

1 

60 

9.489982 

6.48 

9.978206 

.68 

9.511776 

7.17 

10.488224 

0 

' 

Cosine. 

D  r. 

Sine. 

D.  r. 

Cotang. 

D.  r. 

Tang. 

' 

107' 


21 5 


TABLE   XII.      LOGARITHMIC    SINES, 


' 

Sine. 

D.  1". 

Cosine. 

D.  1".    Tang. 

D.  1". 

Cotang. 

t 

0  9.489982 

9.978206 

Aft 

9.511776 

717 

10.488224 

60 

1 

.490371 

a'** 

.978165 

.Do 
Aft 

.512206 

.1< 

.487794 

59 

2 
3 

.490759 
.491147 

O.47 
6.47 

6   Aft 

.978124 

.978083 

.Do 

.68 

Aft 

.512635 
.513064 

7.15 
7.15 

71  f\ 

.487365 
.486936 

58 
57 

4 
5 
6 

.491535 
.491922 

.492308 

.41 

6.45 
6.43 

.978042 
.978001 
.977959 

.Do 

.68 
.70 

.513493 
.513921 
.514349 

.  10 
7.13 
7.13 

.486507 
.486079 
.485651 

56 
55 
54 

7 

.492695 

6.45 

64°. 

.977918 

.68 

Aft 

.514777 

7.13 

.485223 

53 

8 

.493081 

.4o 
649 

.977877 

.DO 

.515204 

7.12 

.484796 

52 

9 

.493466 

.4v* 

6  42 

.977835 

•*** 

.515631 

710 

.484369 

51 

10 

.493851 

6^42 

.977794 

>0 

.516057 

.  1U 

7.12 

.483943 

50 

11 

9.494236 

6  42 

9.977752 

Aft 

9.516484 

•"  in 

10.483516 

49 

12 

.494621 

.977711 

.DO 
7ft 

.516910 

7  ftft 

.483090 

48 

13 

.495005 

6OQ 

.977669 

.  /U 
Aft 

.517335 

(  .Uo 

.482665 

47 

14 
15 
16 

.495388 
.495772 
.496154 

.OO 

6.40 
6.37 

A  °.ft 

.9776.28 
.977586 
.977544 

.Do 

.70 

.70 

Aft 

.517761 

.518186 
.518610 

7'.08 
7.07 

7O7 

.482239 
.481814 
.481390 

46 
45 
44 

17 

18 

.496537 
.496919 

D.oo 

6.37 
607 

.977503 
.977461 

.Uo 

.70 

r-ft 

.519034 
.519458 

.Ui 

7.07 

.480966 
.480542 

43 
42 

19 

.497301 

.01 
6  35 

.977419 

.  <u 

.519882 

7  05 

.480118 

41 

20 

.497682 

6^35 

.977377 

SO 

.520305 

7^05 

.479695 

40 

21 

9.498064 

6oq 

9.977335 

9.520728 

7  OK 

10.479272 

39 

22 

.498444 

.00 

6  OK 

.977293 

''••ft 

.521151 

I  .UO 

.478849 

38 

23 

.498825 

.oO 
609 

.977251 

70 

.521573 

^  ftQ 

.478427 

37 

24 

.499204 

.&* 
600 

.977209 

.  <U 
r/ft 

.521995 

r-  fto 

.478005 

36 

25 
26 

.499584 
.499963 

.  OO 

6.32 

.977167 
.977125 

.  <U 

.70 

.522417 

.522838 

^  .Uo 

7.02 

.477583 
.477162 

35 
34 

27 

.500342 

6.  32 

.977083 

.70 

rft 

.523259 

7.02 

.476741 

33 

28 

.500721 

/>  '  OA 

.977041 

.  tO 

.523680 

r/AA 

.476320 

32 

29 

.501099 

b.oO 

69ft 

.976999 

.70 

7ft 

.524100 

i  .00 

7  ftft 

.475900 

31 

30 

.501476 

./•CO 

6.30 

.976957 

.  <V 

.72 

.524520 

(  .UU 

7.00 

.475480 

30 

31 

9.501854 

69ft 

9.976914 

r'ft 

9.524940 

6  no 

10.475060 

29 

32 

.502231 

.160 

697 

.976872 

.  •  U 

7O 

.525359 

.  Jo 
6  no 

.474641 

2S 

33 
34 

.502607 
.502984 

.35? 

6.28 
fi  27 

.976830 

.976787 

.  <U 

.72 

ry\ 

.525778 
.526197 

.  yo 
6.98 

607 

.474222 
.473803 

27 

26 

35 

.503360 

69K 

.976745 

.  (U 

79 

.526615 

.  y  ( 
6  97 

.473385 

25 

36 
37 

.503735 
.504110 

.60 

6.25 
6  25 

.976702 
.976660 

.  iJB 

.70 

72 

.527033 
.527451 

6^97 

6  OK 

.472967 
.472549 

24 
23 

38 

.504485 

6  "OK 

.976617 

.527868 

.  yo 

.472132 

22 

39 

.504860 

.&> 
69°. 

.976574 

.72 

.528285 

6  95     -471715 

21 

40 

.505234 

.160 

6.23 

.976532 

.72 

.528702 

6^95  1   -4?1298 

20 

41 

9.505608 

9.976489 

rv> 

9.529119 

6  no 

10.470881 

19 

42 

.505981 

699 

.976446 

.  tX 

.529535 

.yo 

6  no 

.470465 

18 

43 

.506354 

.£& 
699 

.976404 

'r-o 

.529951 

.  yo 

6  92 

.470049 

17 

44 

.506727 

.291 

.976361 

•'^ 

.530366 

.469634 

16 

45 

.507099 

6  .20 

.976318 

.  <2 

.530781 

fi  OO 

.469219 

15 

46 
47 

48 

.507471 
.507843 
.508214 

6.20 
6.20 
6.18 
61ft 

.976275 
.976232 
.976189 

.72 

.72 

.531196 
.531611 
.532025 

6^92 
6.90 

.468804 
.468389 
.467975 

14 
13 
12 

49 

.508585 

.  lo 
61ft 

.976146 

'r-o 

.532439 

6  90 

.467561 

11 

50 

.508956 

.  Jo 
6.17 

.976103 

!72 

.532853 

6.  '88 

.467147 

10 

51 

9.509326 

9.976060 

9.533266 

6CQ 

10.466734 

9 

52 
53 
54 

.509696 
.510065 
.510434 

6.17 
6.15 
6.15 
61  z* 

.976017 
.975974 
.975930 

.72 

.72 

•r3 

.533679 
.534092 
.534504 

.OO 

6.88 
6.87 

607 

.466321 
.465908 
.465496 

8 
7 
6 

55 
56 
57 

58 

.510803 
.511172 
.511540 
.511907 

.  ID 

6.15 
6.13 
6.12 

.975887 
.975844 
.975800 
.975757 

!72 
.73 

.72 

.534916 
.535328 
.535739 
.536150 

.o< 

6.87 
6.85 
6.85 

.465084 
.464672 
.464261 
.463850 

5 
4 
3 
2 

59 
60 

.512275 
9.512642 

6.13 
6.12, 

.975714 
9.975670 

.72 
.73 

.536561 
9.536972 

6.85 
6.85 

.463439 
10.463028 

1 
0 

' 

Cosine. 

D.  1". 

Sine. 

D.  r. 

Cotang. 

D.  r. 

Tang.  I  ' 

COSINES,    TANGENTS,    AND    COTANGENTS. 


160* 


' 

Sine. 

D.  r. 

Cosine. 

D.  r. 

Tang. 

D.  r. 

\ 
Cotang. 

' 

o 

9.512642 

9.975670 

170 

9.536972 

600 

10.463028 

60 

1 

.513009 

6.12 

.975627 

.  <  ~ 

.537382 

.00 

(•  Oq 

.462618 

59 

2 

.513375 

6.10 

.975583 

rrt 

.537792 

O.OO 

6oq 

.462208 

58 

2  !  .513741 

6.10 

.975539 

.  to 

.538202 

.00 

60k) 

.461798 

57 

4i 
5 

.514107 
.514472 

6.10 
6.08 

.975496 
.975452 

.72 
.73 

net 

.538611 
*  .539020 

.o/* 

6.82 

6  CO 

.461389 
.460980 

56 
55 

6 

7 
8 

.514837 
.515202 
.515566 

6.08 
6.08 
6.07 

.  975408 
.975365 
.975321 

.  1  o 

.72 
.73 

.539429 
.539837 
.540245 

.0/2 

6.80 
6.80 

.460571 
.460163 
.459755 

54 
53 
52 

9 
10 

.515930 
.516294 

6.07 
6.07 
6.05 

.975277 
.975233 

.73 
.73 
.73 

.540653 
.541061 

6.80 
6.80 
6.78 

.459347 
.458939 

51 
50 

11 

12 

9.516657 

.517020 

6.05 

9.975189 
.975145 

.73 

9.541468 
.541875 

6-£8 

10.458532 
.458125 

49 

48 

13 

.517382 

6.03 

.975101 

.  <o 

.542281 

6'  ^G 

.457719 

47 

14 

.517745 

6.05 

.97'5057 

.73 

.542688 

.  40 

.457312 

46 

15 
16 

.518107 
.518468 

6.03 
6.02 

.975013 
.974969 

.73 
.73 

.543094 
.543499 

6.77 
6.75 

677 

.456906 
.456501 

45 
44 

17 

18 
19 
20 

.518829 
.519190 
.519551 
.519911 

6.02 
6.02 
6.02 
6.00 
6.00 

.974925 
.974880 
.974836 
.974792 

.73 
.75 
.73 
.73 
.73 

.543905 
.544310 
.544715 
.545119 

.  (  I 

6.75 
6.75 
6.73 
6.75 

.456095 
.455690 
.455285 
.454881 

43 
42 
41 
40 

21 
22 
23 

9.520271 
.520631 
.520990 

6.00 

5.98 

9.974748 
.974703 
.974659 

.75 
.73 

9.545524 
.545928 
.546331 

6.73 
6.72 

10.454476 
.454072 
.453669 

39 

38 
37 

24 

.521349 

5.98 

.974614 

.75 

.546735 

6.73 

.453265 

36 

25 

.521707 

5.97 

.974570 

.73 

.547138 

6.72 

.452862 

35 

26 

.522066 

5.98 

.974525 

.75 

.547540 

6.70 

.452460 

34 

27 

.522424 

5.97 

.974481 

.73 

.547943 

6.72 

.452057 

33 

28 

.522781 

5.95 

.974436 

.75 

.548345 

6.70 

.451655 

32 

29 
30 

.523138 
.523495 

5.95 
5.95 
5.95 

.974391 
.974347 

.75 
.73 
.75 

.548747 
.549149 

6.70 
6.70 
6.68 

.451253 
.450851 

31 
30 

31 

9.523852 

9.974302 

9.549550 

10.450450 

29 

32 
33 

.524208 
.524564 

5.93 
5.93 

.974257 
.974212 

.75 
.75 

.549951 
.550352 

6.68 
6.68 

.450049 
.449648 

28 
27 

34 
35 

.524920 
.525275 

5.93 
5.92 

.974167 
.974122 

.75 
.75 

.550752 
.551153 

6.67 
6.68 

.449248  !  26 
.448847  1  25 

36 

.525630 

5.92 

.974077 

.75 

.551552 

6.65 

.448448 

24 

37 

.525984 

5.90 

.974032 

.75 

.551952 

6.67 

.448048 

23 

38 
39 
40 

.526339 
.526693 
.527046 

5.92 
5.90 
5.88 
5.90 

.973987 
.973942 
.973897 

.75 
.75 
.75 
.75 

.552351 
.552750 
.553149 

6.65 
6.65 
6.65 
6.65 

.447649 
.447250 
.446851 

22 
21 

20 

41 

9.527400 

5Oft 

9.973852 

9.553548 

6RQ 

10.446452 

19 

42 
43 

.587753 

.528105 

.00 
5.87 

.973807 
.973761 

.75 

.77 

.553946 
.554344 

.Do 

6.63 

.446054 
.445656 

18 
17 

44 
45 

.528458 
.528810 

5.88 
5.87 

.973716 
.973671 

.75 
.75 

.554741 
.555139 

6.62 
6.63 

.445259 
.444861 

16 
15 

46 
47 

.529161 
.529513 

5.85 
5.87 

.973625 
.973580 

.77 
.75 

.55r«536 
.555933 

6.62 
6.62 

.444464 
.444067 

14 
13 

48 

.529864 

5.85 

.973535 

.75 

.556329 

6.60 

.443671 

12 

49 
50 

.530215 
.530565 

5.85 
5.83 
5.83 

.973489 
.973444 

.77 
.75 

.77 

.556725 
.557121 

6.60 
6.60 
6.60 

.443275 
.442879 

11 
10 

51 
52 

9.530915 
.531265 

5.83 

9.973398 
.973352 

.77 

9.557517 
.557913 

6.60 

10.442483 

.442087 

9 

8 

53 
54 
55 
56 
57 

.531614 
.531963 
.532312 
.532661 
.533009 

5.82 
5.82 
5.82 
5.82 
5.80 

.973307 
.973261 
.973215 
.973169 
.973124 

.75 

.77 
.77 
.77 
.75 

.558308 
.558703 
.559097 
.559491 

.559885 

6.58 
6.58 
6.57 
6.57 
6.57 

.441692 
.441297 
.440903 
.440509 
.440115 

7 
6 
5 
4 
3 

58 
59 

.533357 
.533704 

5.80 

5.78 

.973078 
.973032 

.77 

.77 

.560279 
.560673 

6.57 
6.57 

.439721 
.439327 

2 

1 

60 

9.534052 

5.80 

9.972986 

.77 

9.561066 

6.55 

10.438934 

0 

' 

Cosine. 

D.  r. 

1  Sine.   D.  1*. 

Cotang. 

D.  1'. 

Tang.  1  ' 

109» 


70* 


20° 


TABLE   XII.       LOGARITHMIC    SINES, 


159* 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.l". 

Cotang. 

' 

0 

1 

2 
3 

9.534052 
.534399 
.534745 
.535092 

5.78 
5.77 
5.78 

Siyry 

9.972986 
.972940 
.972894 
.972848 

.77 
.77 
.77 

77 

9.561066 
.561459 
.561851 
.562244 

6.55 
6.53 
6.55 

6cq 

10.438934 
.438541 
.438149 
.437756 

60 
59 
58 
57 

4 
5 
6 

8 
9 
10 

.535438 
.535783 
.536129 
.536474 
.536818 
.537163 
.537507 

.  (  I 

5.75 
5.77 
5.75 
5.73 
5.75 
5.73 
5.73 

.972802 
.972755 
.972709 
.972663 
.972617 
.972570 
.972524 

.  1  1 

.78 

.77 
.77 
.77 
.78 
.77 
.77 

.562636 
.563028 
.563419 
.563811 
.564202 
.564593 
.564983 

.Oo 

6.53 
6.52 
6.53 
6.52 
6.52 
6.50 
6.50 

.437364 
.436972 
.436581 
.436180 
.435798 
.435407 
.435017 

56 
55 
54 
53 
52 
51 
50 

11 
12 
13 
14 

9.537851 
.538194 
.538538 

.538880 

5.72 

5.73 
5.70 

9.972478 
.972431 
.972385 
.972338 

.78 
.77 
.78 

78 

9.565373 
.565763 
.566153 
.566542 

6.50 
6.50 
6.48 

6CA 

10.434627 
.434237 
.433847 
.433458 

49 
48 
47 
46 

15 
16 
17 
18 
19 
20 

.539223 
.539565 
.539907 
.540249 
.540590 
.540931 

5.72 
5.70 
5.70 
5.68 
5.68 
5.68 
5.68 

.972291 
.972245 
.972198 
.972151 
.972105 
.972058 

.  to 

.77 
.78 
.78 
.77 
.78 
.78 

.566932 
.567320 
.567709 
.568098 
.568486 
.568873 

.OU 

6.47 
6.48 
6.48 
6.47 
6.45 
6.47 

.433068 
.432680 
.432291 
.431902 
.431514 
.431127 

45 
44 
43 
42 
41 
40 

21 

9.541272 

5  no 

9.972011 

r<p 

9.569261 

64R 

10.430739 

39 

22 
23 

.541613 
.541953 

.DO 

5.67 

.971964 
.971917 

.  <O 

.78 

78 

.569648 
.570035 

.40 

6.45 
6  45 

.430352 
.429965 

38 
37 

24 

.542293 

2*2£ 

.971870 

.  <  0 
p»p 

!  .570422 

6AK 

.429578 

36 

25 
26 

27 
28 

.542632 
.542971 
.543310 
.543649 

5.o5 
5.65 
5.65 
5.65 

5RQ 

.971823 
.971776 
.971729 
.971682 

.  to 

.78 
.78 
.78 

!  .570809 
.571195 
.571581 
.571967 

.40 
6.43 
6.43 
6.43 
6  42 

.429191 
.428805 
.428419 
.428033 

35 
34 
33 
32 

29 

.543987 

.Do 

.971635 

'Xo 

.572352 

.427648 

31 

30 

.544325 

5.63 
5.63 

.971588 

.  to 
.80 

.572738 

6^42 

.427'262 

30 

31 
32 
33 

9.544663 
.545000 
.545338 

5.62 
5.63 

9.971540 
.971493 
.971446 

.78 
.78 
en 

9.573123 
.573507 
.573892 

6.40 
6.42 

10.426877 
.426493 
.426108 

29 

28 
27 

34 

.545674 

5.60 

5RO 

.971398 

.oU 
78 

.574276 

640 

.425724  26 

35 

.546011 

.  \>& 

.971351 

.  to 

.574660 

.4U 

.425340 

25 

36 

.546347 

5.60 

.971303 

.80 

78 

.575044 

6qp 

.424956 

24 

37 

.546683 

5.60 

.971256 

.  to 

.575427 

.OO 

6qp 

.424573 

23 

38 
39 
40 

.547019 
.547354 
.547689 

5.60 
5.58 
5.58 
5.58 

.971208 
.971161 
.971113 

.80 
.78 
.80 

.78 

.575810 
.576193 
.576576 

.oO 

6.38 
6.38 
6.38 

.424190 
.423807 
.423424 

22 
21 

20 

41 

9.548024 

9.971066 

PA. 

9.576959 

6q7 

10.423041 

19 

42 

.548359 

5.58 

.971018 

.oU 

.577341 

.04 

.422659 

18 

43 

44 

.548693 
.549027 

5.57 
5.57 

.970970 
.970922 

.80 

.80 
p/i 

.577723 
.578104 

6.37 
6.35 

6q7 

.422277 
.421896 

17 
16 

45 

46 

.549360 
.549693 

5.55 
5.55 

5KK 

.970874 
.970827 

.OU 

.78 

PA 

.578486 
.578867 

.01 

6.35 
6  35 

.421514 
.421133 

15 
14 

47 

48 

.550026 
.550359 

.OO 

5.55 

5KK 

.970779 
.970731 

.OU 

.80 

PA 

.579248 
.579629 

6^35 

600 

.420752 
.420371 

13 
12 

49 

.550692 

.OO 

5KQ 

.970683 

.OU 
8A 

.580009 

.00 

6qo 

.419991 

11 

50 

.551024 

.Oo 

5.53 

.970635 

.OU 

.82 

.580389 

.00 

6.33 

.419611 

10 

51 

9.551356 

5feO 

9.970586 

PA 

8.580769 

A  q.q 

10.419231 

9 

52 

.551687 

.O<<J 
5  to 

.970538 

.OU 
PA 

.581149 

D.oo 

6  32 

.418851 

8 

53 
54 

.552018 
.552349 

.0/6 

5.52 

5KO 

.970490 
.970442 

.OU 

.60 

PA 

.581528 
.581907 

6.  '32 

6qo 

.418472 
.418093 

7 
6 

55 

56 
57 
58 

.552680 
.553010 
.553341 
.553670 

,OC3 

5.50 
5.52 
5.48 

5tA 

.970394 
.970345 
.970297 
.970249 

.OU 

.82 
.80 
.80 

.582286 
.582665 
.583044 
.583422 

.06 

6.32 
6.32 
6.30 

6qA 

.417714 
.417335 
.416956 
.416578 

5 
4 
3 
2 

59 
60 

.554000 
9.554329 

.OU 

5.48 

.970200 
9.970152 

.'80 

.583800 
9.584177 

.oU 

6.28 

.416200 
10.415823 

1 
0 

' 

Cosine. 

D.I". 

Sine. 

D.  r. 

Cotang.   D.  1*. 

Tang.  |  ' 

69' 


COSINES,    TANGENTS,    AND    COTANGENTS 
21*  108° 


Sine. 

D.  r. 

Cosine. 

D.  r. 

Tang. 

D.  1". 

Cotang. 

' 

0 

9.554329 

9.97'0152 

9.584177 

6OA 

10.415823 

60 

1 

2 

.554658 
.554987 

5^48 

.970103 
.970055 

!so 

.584555 
.584932 

.  dU   , 

6.28 

.415445 
.415068 

59 
58 

3 

.555315 

5.47 

.970006 

.82 

.585309 

6.28 

.414691 

57 

4 

.555643 

5.47 

.969957 

.82 

.585686 

6.28 

.414314 

56 

5 

.555971 

5.47 

.969909 

.80 

.586062 

6.27 

.413938 

55 

6 

.556299 

5.47 

.969860 

.82 

.586439 

6.28 

.413561 

54 

7 
8 
9 
10 

.556626 
.556953 
.557280 
.557606 

5.45 
5.45 
5.45 
5.43 
5.43 

.969811 
.969762 
.969714 
.969665 

.82 

.82 
.80 
.82 
.82 

.586815 
.587190 
.587566 
.587941 

6.27 
6.25 
6.27 
6.25 
6.25 

.413185 
.412810 
.412434 
.412059 

53 
52 
51 
50 

11 
12 

9.557932 

.558258 

5.43 

9.969616 
.969567 

.82 

9.588316 
.588691 

6.25 

10.411684 
.411309 

49 

48 

13 

.558583 

5.42 

.969518 

.82 

.589066 

6.25 

.410934 

47 

14 

.558909 

O.4d 

.969469 

.82 

.589440 

6.23 

.410560 

46 

15 

.559234 

5.42 

.969420 

.82 

.589814 

6.23 

.410186 

45 

16 

.559558 

5.40 

5A9 

.969370 

.83 

.590188 

6.23 
690 

.409812 

44 

17 

.559883 

.4/<& 

.969321 

.8,4 

.590562 

./GO 

.409438 

43 

18 

.560207 

5.40 

.969272 

.82 

.590935 

6.22 
600 

.409065 

42 

19 

.560531 

t'^A 

.969223 

00 

.591308 

,/sz 

.408692 

41 

20 

.560855 

5.40 
5.38 

.969173 

.83 
.82 

.591681 

6.22. 
6.22 

.408319 

40 

21 

22 

9.561178 
.561501 

5.38 

9.969124 
.969075 

.82 

9.592054 
.592426 

6.20 

10.407946 
.407574 

39 

38 

23 
24 
25 

.561824 
.562146 
.562468 

5.38 
5.37 
5.37 

5qr< 

.969025 
.968976 
.968926 

.83 
.82 
.83 

.592799 
.593171 
.593542 

6.22 
6.20 
6.18 

.407201 
.406829 
.406458 

37 
36 
35 

26 

.562790 

.of 

.968877 

.82 

.593914 

6.20 

.406086 

34 

27 

.563112 

5.37 

.968827 

.83 

.594285 

6.18 

.405715 

33 

28 

.563433 

5.35 

.968777 

.83 

.594656 

6.18 

.405344 

32 

29 

.563755 

5.37 

.968728 

.82 

.595027 

6.18 

.404973 

31 

30 

.564075 

5.33 
5.35 

.968678 

.83 
.83 

.595398 

6.18 
6.17 

.404602 

30 

31 

9.564396 

5QQ 

9.968628 

9.595768 

10.404232 

29 

32 

.564716 

.dO 
5qq 

.968578 

.83 

.596138 

D.17 
64  m 

.403862 

28 

33 

.565036 

.dO 

.968528 

.83 

.596508 

.if 

.403492 

27 

34 

.565356 

5.33 
5qq 

.968479 

.82 

.596878 

6.17 
61K 

.403122 

26 

35 

.565676 

.OO 

.968429 

.83 

.567247 

.  lo 

.402753 

25 

36 

.565995 

5.32 

.968379 

.83 

.597616 

6.15 

.402384 

24 

37 

.566314 

5.32 

.968329 

.83 

.597985 

6.15 

.402015 

23 

38 
39 

.566632 
.566951 

5.30 
5.32 

.968278 
.968228 

.85 

.83 

.598354 
.598722 

6.15 
6.13 

.401646 
.401278 

22 

21 

40 

.567269 

5.30 
5.30 

.968178 

.83 
.83 

.599C91 

6.15 
6.13 

.400909 

20 

41 

9.567587 

9.968128 

9.599459 

61Q 

10.400541 

19 

42 
43 

.567904 
.568222 

5^30 

.968078 
.968027 

.83 

.85 

.599827 
.600194 

.id 

6.12 

.400173 
.399806 

18 
17 

44 

.568539 

5.28 

.967977 

.83 

.600562 

6.13 

.399438 

1ft 

45 
46 

.568856 
.569172 

5.28 
5.27 

.967927 
.967876 

.83 
.85 

.600929 
.601296 

6.12 
6.12 

.399071 
.398704 

15 
14 

47 
48 

.569488 
.569804 

5.27 
5.27 
5.27 

.967826 
.967775 

.83 
.85 

Qq 

.601663 
.602029 

6.12 
6.10 

.398337 
.397971 

13 
12 

49 
50 

.570120 
.570435 

5.25 
5.27 

.967725 
.967674 

.OO 

.85 
.83 

.602395 
.602761 

6.'  10 
6.10 

.397605 
.397239 

11 
10 

51 

52 
53 
54 

9.570751 
.571066 
.571380 
.571695 

5.25 
5.23 
5.25 

9.967624 
.967573 
.967522 
.967471 

.85 
.85 
.85 

9.603127 
.603493 
.603858 
.604223 

6.10 

6.08 
6.08 

10.396873 
.396507 
.396142 
.395777 

9 
8 

7 
5 

55 

.572009 

5.23 

.967421 

.83 

.604588 

6.08 

.395412 

5 

56 

.572323 

5.23 

.967370 

.85 

.604953 

6.08 

.395047 

4 

57 

.572636 

5.22 

.967319 

.85 

.605317 

6.07 

.394683 

3 

58 

.572950 

5.23 

.967268 

.85 

.605682 

6.08 

.394318 

2 

59 
60 

.573263 
9.573575 

5.22 
5.20 

.967217 
9.967166 

.85 

.85 

.606046 
9.606410 

6.07 
6.07 

.393954 
10.393590 

1 

0 

' 

Cosine. 

D.  1".  1  1  Sine. 

D.  1'. 

Cotang.  1  D.  1".  1  Tang. 

' 

111- 


68' 


22* 


TABLE  XII.      LOGARITHMIC   SINES, 


' 

Sine. 

D.  1'. 

Cosine. 

D.  I". 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

2 
3 

4 

9.573575 

.573888 
.574200 
.574512 

.574824 

5.22 

5.20 
5.20 
5.20 

9.967166 
.967115 
.967'064 
.967013 
.966961 

.85 
.85 
.85 

.87 

9.606410 
.606773 
.607137 
.607500 
.607863 

6.05 
6.07 
6.05 
6.05 

10.393590 
.393227 
.392863 
.392500 
.392137 

60 

59 
58 

57 
56 

5 

.575136 

K'-IQ 

.966910 

.85 

OK 

.608225 

6.03 

.391775 

55 

6 

.575447 

O.  io 

5  18 

.966859 

.oO 

OK 

.608588 

6.05 

6AO 

.391412 

54 

7 
8 

.575758 
.576069 

5!l8 

517 

.966808 
.966756 

.oO 

.87 

.608950 
.609312 

.Uo 
6.03 

.391050 

.390688 

53 
52 

9 
10 

.576379 
.576689 

.If 

5.17 
5.17 

.966705 
.966653 

.85 
.87 
.85 

.609674 
.610036 

6.03 
6.03 
6.02 

.390326 
.389964 

51 

50 

11 
12 

9.576999 
.577309 

5.17 

5-iK 

9.966602 
.966550 

.87 

OK 

9.610397 
.610759 

6.03 

10.389603 
.389241 

49 

48 

13 

14 
15 
16 
17 
18 
19 
20 

.577618 
.577927 
.578236 
.578545 
.578853 
.579162 
.579470 
.579777 

.  10 
5.15 
5.15 
5.15 
5.13 
5.15 
5.13 
5.12 
5.13 

.966499 
.966447 
.966395 
.966344 
.966292 
.966240 
.966188 
.966136 

.oO 

.87 
.87 
.85 
.87 
.87 
.87 
.87 
.85 

.611120 
.611480 
.611841 
.612201 
.612561 
.612921 
.613281 
.613641 

6^00 
6.02 
6.00 
6.00 
6.00 
6.('0 
6.00 
5.98 

.888880 
.388520 
.388159 
.387799 
.387439 
.387079 
.386719 
.386359 

47 
46 
45 
44 
43 
42 
41 
40 

21 
22 

9.580085 
.580392 

5.12 

9.966085 
.966033 

.87 

9.614000 
.614359 

5.98 

10.386000 
.385641 

39 

38 

23 
24 
25 
26 
27 

.580699 
.581005 
.581312 
.581618 
.581924 

5.  12 
5.10 
5.12 
5.10 
5.10 

.965981 
.965929 
.965876 
.965824 
.965772 

.87 
.87 
.88 
.87 
.87 

.614718 
.615077 
.615435 
.615793 
.616151 

5.98 
5.98 
5.97 
5.97 
5.9V 

.385282 
.384923 
.384565 
.384207 
.383849 

37 
36 
35 
34 
33 

28 
29 

.582229 
.582535 

5  '.10 

5AQ 

.965720 
.965668 

.87 
.87 

00 

.616509 
.616867 

5.97 
5.97 

5QK 

.383491 
.383133 

32 
31 

30 

.582840 

.Uo 

5.08 

.965615 

.00 
.87 

.617224 

.  yo 
5.97 

.382776 

30 

31 
32 

9.583145 
.583449 

5.07 

5  AD 

9.965563 
.965511 

.87 

QQ 

9.617582 
.617939 

5.95 

5  no 

10.382418 
.382061 

20 

28 

33 
34 

.583754 
.584058 

.Uo 

5.07 

5AK 

.965458 
.965406 

.OO 

.87 

QQ 

.618295 
.918652 

.  yo 
5.95 

.381705 
.381348 

27 
26 

35 

.584361 

.uo 

5A7 

.965353 

.OO 

8r» 

.619008 

5.93 

5  no 

.380992 

25 

36 

.584665 

.Uf 

5  05 

.965301 

t 

QQ 

.619364 

.yo 

5Q°> 

.380636 

24 

37 

.584968 

5A.7 

.965248 

.OO 
QQ 

.619720 

.yo 

5  no 

.380280 

23 

38 
39 

.585272 
.585574 

.u< 
5.03 

5  A|* 

.965195 
.965143 

.OO 

.87 

QQ 

.620076 
.620432 

.y»j 
5.93 

.379924 
.379568 

22 

21 

40 

.585877 

.uo 
5.03 

.965090 

.OO 

.88 

.620787 

5.92 
5.92 

.379213 

20 

41 

9.586179 

5  OK 

9.965037 

QQ 

9.621142 

10.378858 

19 

42 

.586482 

.UO 

.964984 

.OO 

.621497 

^'n^ 

.378503 

18 

43 

.586783 

5.02 

5AO 

.964931 

.88 

fi.7 

.621852 

5.92 

.378148 

17 

44 

45 
46 

.587085 
.587386 
.587688 

.uo 
5.02 
5.03 

.964879 
.964826 
.964773 

.Of 

.88 
.88 

.622207 
.622561 
.622915 

5.  '90 
5.90 

.377793 
.377439 
.377085 

16 
15 
14 

47 
48 
49 

.587989 
.588289 
.588590 

5.02 
5.00 
5.02 
5nn 

.964720 
.964666 
.964613 

.88 
.90 
.88 

QQ 

.623269 
.623623 
.623976 

5.90 
5.90 

5.88 

5QA 

.376731 
.376377 
.376024 

13 

12 
11 

50 

.588890 

.uu 
5.00 

.964560 

.OO 

.88 

.624330 

.  yu 

5.88 

.375670 

10 

51 

9.589190 

9.964507 

9.624683 

10.375317 

9 

52 
53 

.589489 
.589789 

4.98 
5.00 

.964454 
.964400 

.88 
.90 

.625036 

.625388 

5.88 
5.87 

.374964 
.374612 

8 

54 

.590088 

4.98 

.964347 

.88 

.625741 

5.88 

.374259 

6 

55 

.590387 

4.98 

4QQ 

.964294 

.88 

.626093 

5.87 

5C7 

.373907 

5 

56 

.590686 

.yo 

.964240 

.90 

.626445 

.o< 

.373555 

4 

57 

58 

.590984 
.591282 

4.97 
4.97 

.964187 
.964133 

.88 
.90 

.626797 
.627149 

5.87 
5.87 

.373203 
.372851 

3 
2 

59 
60 

.591580 
9.591878 

4.97 
4.97 

.964080 
9.964026 

.88 
.90 

.627501 
9.627852 

5.87 
5.85 

.372499 
10.372148 

1 
0 

' 

Cosine.  |  D.  1". 

Sine. 

D.  1*. 

Cotang. 

D.  r. 

Tang.  |  ' 

220 


COSINES,   TANGENTS.   AND   COTANGENTVS. 


' 

Sine. 

D.  1'. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

9.591878 
.592176 

4.97 

9.964026 
.963972 

.90 

ftft 

9.627852 
.628203 

5.85 

5QK 

10.372148 
.371797 

60 
59 

2 
3 

4 
5 

.592473 
.592770 
.593067 
.593363 

4.95 
4.95 
4.95 
4.93 

4QO 

.963919 
.963865 
.963811 
.963757 

.00 
.90 
.90 
.90 

ftR 

.628554 
.628905 
.629255 
.629606 

.OO 

5.85 
5.83 

5.85 

500 

.371446 
.371095 
.370745 
.370394 

58 
57 
56 
55 

6 

.593659 

.yo 

.963704 

.OO 

.629956 

.00 

K  pq 

.370044 

54 

7 
8 
9 

.593955 
.594251 
.594547 

4^93 
4.93 

.963650 
.963596 
.963542 

'.90 
.90 

Qrt 

.630306 
.630656 
.631005 

O.oo 

5.83 

5.82 

5pq 

.369C94 
.369344 
.368995 

53 
52 
51 

10 

.594842 

4.92 
4.92 

.963488 

.yu 
.90 

.631355 

.Oo 

5.82 

.368645 

50 

11 

9.595137 

4Q9 

9.963434 

no 

9.631704 

10.368296 

49 

12 

.595432 

.M 

.963379 

.632053 

r  po 

.367947 

48 

13 

.595727 

4.  92 

.963325 

*Qrt 

.632402 

5ftO 

.367598 

47 

14 
15 
16 

.596021 
.596315 

.596609 

4!90 
4.90 

.963271 
.963217 
.963163 

.yu 
.90 
.90 

.632750 
.633099 
.633447 

.oU 
5.82 
5.80 

K  pn 

.367250 
.366901 
.366553 

46 
45 
44 

17 

.596903  1  ?'™ 

.963108 

Qrt 

.633795 

o.ou 

K  PA 

.366205 

43 

18 

.597196   J'JS 

.963054 

.yu 

.634143 

O  .  OU 

.365857 

42 

19 

.597490  1  2'ftft 

.962999 

*Qrt 

.634490 

5prt 

.365510 

41 

20 

.597783   |'gy 

.962945 

.yu 
.92 

1  .634838 

.oU 

5.78 

.3fi5162 

40 

21 

22 
23 
24 

9.598075 
.598368 
.598660 
.598952 

4.88 

4.87 
4.87. 

0.962890 
.962836 
.962781 
.962727 

.90 
.92 
.90 

9.635185 
.635532 
.635879 
.636226 

5.78 
5.78 
5.78 

10.364815 
.364468 
.364121 
.363774 

39 
38 
37 
36 

2a 

.599244 

4.o« 
4P.7 

.962672 

.92 

.636572 

5.77 

.363428 

35 

26 

.599536 

.Of 

4QK 

.962617 

QO 

.636919 

K-77 

.363081 

34 

27 

28 

.599827 
.600118 

.00 

4.85 

.962562 
.962503 

!90 

.637265 
.637611 

O.It 

5.77 

.362735 

.362389 

33 
32 

29 
30 

.600409 
.600700 

4.85 
4.85 
4.83 

.962453 
.962398 

.92 
.92 
.92 

.637956 
.638302 

5.75 
5.77 
5.75 

.362044 
.361698 

31 
30 

31 

9.600990 

4pq 

9.962343 

QO 

9.638647 

5  75 

10.361353 

29 

32 

.601280 

.  oo 

.962288 

.MB 

.638992 

.361008 

28 

33 

.601570 

4.83 

.962233 

.92 

.639337 

5.75 

.360663 

27 

34 

.601860 

4.83 

.962178 

.92 

.639682 

5.75 

.360318 

26 

35 

.602150 

4.83 

.962123 

.92 

.640027 

5.75 

57°. 

.359973 

25 

36 
37 

.602439 
.602728 

4^82 

4QO 

.962067 
.962012 

!92 

.640371 
.640716 

.  !•) 

5.75 

570 

.359629 
.359284 

24 

23 

33 

.603017 

.0.* 

.961957 

.92 

.641060 

.  to 

.358940 

22 

39 

.603305 

4.80 

.961902 

.92 

.641404 

5.73 

579 

.358596 

21 

40 

.603594 

4^80 

.961846 

!92 

.641747 

.  1  4 

5.73 

.358253 

20 

41 

9.603883 

4  OA 

9.961791 

Q°, 

9  642091 

579 

10.357909 

19 

42 

.604170 

.OU 

47ft 

.961735 

Q9 

.642434 

.  <  V 

579 

.357566 

10 

43 

.604457 

.  to 

4  QA 

.901680 

.MB 

QO 

.642777 

.  <  ~/ 
679 

.357223 

17 

44 

.604745 

s  OU 

.961624 

,yo 

.643120 

.  tfi 

.356880 

16 

45 

.605032 

4.78 

.961569 

.92 

no 

.643463 

5.72 

579 

.356537 

15 

45 

47 
48 

.605319 
.605606 
.605892 

4.78 

4.77 

.961513 
.961458 
.961402 

.yo 
.92 
.93 

.643806 
.644148 
.644490 

.  (  V 

5.7'0 
5'J° 

.356194 
.355852 
.355510 

14 
13 
12 

49 
50 

.603179 
.606465 

4.  77 
4.77 

.961346 
.961290 

!93 
.92 

.644832 
.645174 

5!  70 
5.70 

.355168 
.354826 

11 

10 

51 

9.606751 

4^K 

9.961235 

9.645516 

500 

10.354484 

9 

52 

.607036 

.  <o 

4.  1  1 

.961179 

'no 

.645857 

.  Oo 

.354143 

8 

53 

.607322 

.961123 

.I/O 

!  .646199 

*'  n 

.353801 

7 

54 

.607607 

4.75 

.961067 

.93 

i  .646540 

5.68 

.353460 

6 

55 

.607892 

4.75 

.961011 

.93 

.646881 

5.68 

.353119 

5 

56 

.608177 

4.75 

.960955 

.93 

.647222 

5.68 

.352778 

4 

57 
58 

.608461 
.608745 

4.73 
4.73 
470 

.960899 
.960843 

.93 
.93 

.647562 
.647903 

5.67 
5.68 

.352438 
.352097 

3 

2 

59 
60 

.609029 
9.609313 

.  <o 

4.73 

.960786 
9.960730 

'.93 

.648243 
9.648583 

5.67 
5.67 

.351757 
10.351417 

1 
0 

'  I  Cosine. 

D.  1". 

Sine. 

D.  1". 

i  Cotang. 

D.  1*. 

Tang. 

' 

60° 


TABLE   XII.       LOGARITHMIC    SINES., 


, 

' 

Sine. 

D.I". 

Cosine. 

D.  1". 

Tang. 

D.  1".   Cotang. 

' 

0  9.609313 

9.960730 

QO. 

9.648583 

10.351417 

60 

1 
2 
3 
4 
5 

.609597 
.609880 
.610164 
.610447 
.610729 

4^72 
4.73 
4.72 
4.70 

.960674 
.960618 
.960561 
.960505 
.960448 

.yo 
.93 
.95 
.93 
.95 

.648923 
.649263 
.649602 
.649942 
.650281 

5.67 
5.67 
5.65 
5.67 
5.65 

.351077 
.350737 
.350398 
.350058 
.349719 

59 
58 
57 
56 
55 

6 

7 

.611012 
.611294 

4.72 
4.70 
4  70 

.960392 
.960335 

.93 
.95 

QO 

.650620 
.650959 

5.65 
5.65 

.349380  i  54 
.3.49041  53 

8 
9 

.611576 
.611858 

4.  70 
4  70 

.960279 
.960222  } 

.yo 
.95 

QK 

.651297 
.651636 

5^65 

5  on 

.348703  52 
.348364  51 

10 

.612140 

4^68 

.960165 

.yo 

.93 

.651974 

.Do 

5.63 

.348026 

50 

11 

9.612421 

A  AQ 

9.960109 

95 

9.652312 

5CQ 

10.347688 

49 

12 
13 
14 

.612702 
.612983 
.613264 

4.  Oo 

4.68 
4.68 

4  PA 

.960052 
.959995 
.959938 

!95 
.95 

.652650 
.652988 
.653326 

.Do 

5.63 
5.63 

.347350  48 
.347012  i  47 
.346674  !  46 

15 
16 

.613545 
.613825 

.Do 

4.67 

4fi7 

.959882 
.959825 

!95 

QK 

.653663 
.654000 

5.62 
5.62 

5  en 

.346337  45 
.346000  i  44 

17 
18 
19 

.614105 
.614385 
.614665 

.Of 

4.67 
4.67 

.959768 
.959711 
.959654 

.yo 

.95 

.95 

9.7 

.654337 
.654674 
.655011 

.  <  >  si 

5.62 
5.62 

.345663 
.345326 
.344989 

43 
42 
41 

20 

.614944 

4.65 
4.65 

.959596 

i 
.95 

.655348 

5.62 
5.60 

.344652 

40 

21 
22 
23 
24 
25 
26 
27 

9.615223 
.615502 
.615781 
.616060 
.616338 
.616616 
.616894 

4.65 
4.65 
4.65 
4.63 
4.63 
4.63 

4  on 

9.959539 
.959482 
.959425 
.959368 
.959310 
.959253 
.959195 

.95 
.95 
.95 
.97 
.95 
.97 

QK 

9.  655684  „ 
.656020 
.656356 
.656692 
.657028 
.657364 
.657699 

5.60 
5.60 
5.60 
5.60 
5.60 
5.58 

5KQ 

10.344316 
.343980 
.343644 
.343308 
.342972 
.342636 
.342301 

39 
38 
37 
36 
35 
34 
33 

28 

.617172 

.Do 

4  on 

.959138 

.yo 

Q7 

.658034 

.Oo 

K  KQ 

.341966 

32 

29 

.617450 

.  Do 

.959080 

.y< 

QK 

.658369 

o  .00 

5KQ 

.341631 

31 

30 

.617727 

4.62 
4.62 

.959023 

.yo 

.97 

.658704 

.  Oo 

5.58 

.341296 

30 

31 

9.618004 

9.958965 

QK 

9.659039 

10.340961 

29 

32 
33 
34 

.61&81 

.618558 
.618834 

4^62 
4.60 

.958908 
.958850 
.958792 

.yo 
]97 

.659373 
.659708 
.660042 

5^58 
5.57 

5K/7 

.340627 
.340292 
.339958 

28 
27 
26 

35 
36 
37 

38 
39 
40 

.619110 
.619386 
.619662 
.619938 
.620213 
.620488 

4  '.60 
4.60 
4.60 
4.58 
4.58 
4.58 

.958734 
.958677 
.958619 
.958561 
.958503 
.958445 

'.95 

.97 
.97 
.97 
.97 
.97 

.660376 
.660710 
.661043 
.661377 
.661710 
.662043 

»OV 

5.57 
5.55 
5.57 
5.55 
5.55 
5.55 

.339624 

!  338957 
.338623 
.338290 
.337957 

25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 

9.620763 
.621038 
.621313 
.621587 
.621861 
.622135 
.622409 

4.58 
4.58 
4.57 
4.57 
4.57 
4.57 

4KK 

9.958387 
.958329 
.958271 
.958213 
.958154 
.958096 
.958038 

.97 
.97 
.97 
.98 
.97 
.97 

QQ 

9.662376 
.662709 
.663042 
.6b3375 
.663707 
.664039 
.664371 

5.55 
5.55 
5.55 
5.53 
5.53 
5.53 

5KO 

10.337624 
.337291 
,336958 
.336625 
.336293 
.335961 
.335629 

19 
18 
17 
16 
15 
14 
13 

48 
49 
50 

.622682 
.622956 
.623229 

.00 

4.57 
4.55 
4.55 

.957979 
.957921 
.957863 

.yo 
.97 
.97 
.98 

.664703 
.665035 
.665366 

.Oo 

5.53 
5.52 
5.53 

.335297 
.3349G5 
.334634 

12 
11 
10 

51 

52 

9.623502 
.623774 

4.53 

9.957804 
.957746 

.97 

QQ 

9.665698 
.666029 

5.52 

10.334302 
.333971 

9 

8 

53 

.624047 

4.55 

.957687 

.yo 

.666360 

5.52 

5KO 

.333640 

7 

54 
55 
56 

.624319 
.624591 
.624863 

4.53 
4.53 
4.53 

.957628 
.957570 
.957511 

!97 
.98 

.666691 
.667021 
.667352 

.0.* 

5.50 
5.52 

.333309 
.332979 
.332648 

6 
5 

4 

57 
58 
59 
60 

.625135 
.625406 
.625677 
9.625948 

4.53 
4.52 
4.52 
4.52 

.957452 
.957393 
.957335 
9.957276 

.98 
.98 
.97 
.98 

.667682 
.668013 
.668343 
9.668673 

5.50 
5.52 
5.50 
5.50 

.332318 
.331987 
.331657 
10.331327 

3 

2 

1 
0 

' 

Cosine.  1  D.  1*. 

Sine, 

D.I". 

Cotang. 

D.  1". 

Tang. 

' 

25° 


COSINES,    TANGENTS,    AND   COTANGENTS. 


' 

Sine. 

D.  r. 

Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

' 

0 

9.625948 

9.95T276 

QQ 

9.668673 

540 

10.331327 

60 

1 

.626219 

At-n 

.957217 

.yo 

.669002 

.10 

.330998 

59 

2 

.626490 

4.5/6 

.957158 

'Xo 

.669332 

5.50 

.330668 

58 

3 

.626760 

4.50 

.957099 

.98 

no 

.669661 

o.48 

5tA 

.330339 

57 

4 

.627030 

4.0U 

.957040    •£? 

.669991 

.ou 

.330009 

56 

5 
6 

7 

.627300 
.627570 
.627840 

4.50 
4.50 
4.50 

.956981 
.956921 
.956862 

.yo 
1.00 
.98 

QQ 

.670320 
.670649 
.670977 

5.48 
5.48 
5.47 

K  AQ 

.329680 
.329351 
.329023 

55 
54 
53 

8 

.628109 

4.48 

.956803 

.yo 

.671306 

O.40 

.328694 

52 

9 
10 

.628378 
.628647 

4.48 
4.48 
4.48 

.956744 
.956684 

.98 
1.00 
.98 

.671635 
.671963 

5.48 
5.47 
5.47 

.328365 
.328037 

51 
50 

11 

9.628916 

9.956625 

QQ 

9.672291 

10.327709 

49 

12 
13 

.629185 
.629453 

4^47 

.956566 
.956506 

.yo 
1.00 

QQ 

.672619 
.672947 

5.47 
5.47 

5AK 

.327381 
.327053 

48 

47 

14 
15 
16 

.629721 
.629989 
.630257 

4.47 
4.47 
4.47 

.956447 
.956387 
.956327 

.yo 
1.00 
1.00 

QQ 

.673274 
.673602 
.673929 

.40 

5.47 
5.45 

.326726 
.326398 
.326071 

46 
45 
44 

17 
18 
19 

.630524 
.630792 
.631059 

4.45 
4.47 
4.45 

.956268 
.056208 
.956148 

.yo 
1.00 
1.00 

.674257 
.674584 
.674911 

5.47 
5.45 
5.45 

.325743 
.325416 
.325089 

43 
42 
41 

20 

.631326 

4.45 
4.45 

.956089 

.98 

i.oo 

.675237 

5.43 
5.45 

.324763 

40 

21 
22 
23 
24 
25 
26 
27 
28 

9.631593 
.631859 
.632125 
.632392 
.632658 
.632923 
.633189 
.633454 

4.43 
4.43 
4.45 
4.43 
4.42 
4.43 
4.42 

9.956029 
.955969 
.955909 
.955849 
.955789 
.955729 
.955669 
.955609 

1.00 
1.00 
1.00 
1.00 
1.00 
1.00 
1.00 

9.675564 
.675890 
.676217 
.676543 
.676869 
.677194 
.677520 
.677846 

5.43 
5.45 
5.43 
5.43 
5.42 
5.43 
5.43 

10.324436 
.324110 
.323783 
.323457 
.323131 
.322806 
.322480 
.322154 

39 
38 
37 
36 
35 
34 
33 
32 

29 
30 

.633719 
.633984 

4.42 
4.42 
4.42 

.955548 
.955488 

.98 
1.00 
1.00 

.678171 
.678496 

5.42 
5.42 
5.42 

.321829 
.321504 

31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.634249 
.634514 
.634778 
.635042 
.635306 
.635570 
.035834 
.636097 
.636360 
.636623 

4.42 
4.40 
4.40 
4.40 
4.40 
4.40 
4.38 
4  38 
4.38 
4.38 

9.955428 
.955368 
.955307 
.955247 
.955186 
.955126 
.955065 
.955005 
.954944 
.954883 

1.00 
1.02 
1.00 
1.02 
1.00 
1.02 
1.00 
1.02 
1.02 
1.00 

9.678821 
.679146 
.679471 
.679795 
.680120 
.680444 
.680768 
.681092 
.681416 
.681740 

5.42 
5.42 
5.40 
5.42 
5.40 
5.40 
5.40 
5.40 
5.40 
5.38 

10.321179 
.320854 
.320529 
.320205 
.319880 
.319556 
.319232 
.318908 
.318584 
.318260 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 

44 
45 
46 
47 
48 
49 
50 

9.636886 
.637148 
.637411 
.637673 
.637935 
.638197 
.638458 
.638720 
.638981 
.639242 

4.37 
4.38 
4.37 
4.37 
4.37 
4.35 
4.37 
4.35 
4.35 

9.954823 
.954762 
.954701 
.954640 
.954579 
.954518 
.954457 
.954396 
.954335 
.954274 

1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 

9.682063 
.682387 
.682710 
.683033 
.683356 
.683679 
.684001 
.684324 
.684646 
.684968 

5.40 
5.38 
5.38 
5.38 
5.38 
5.37 
5.38 
5.37 
5.37 
507 

10.317937 
.317613 
.317290 
.316967 
.316644 
.316321 
.315999 
.315676 
.315354 
.315032 

19 

18 

11 
It 

13 
12 
11 
10 

4.35 

1.02 

.01 

51 
52 

9.639503 
.639764 

4.35 

9.954213 
.954152 

1.02 

9.685290 
.685612 

5.37 

10.314710 
.314388 

9 

8 

53 

.640024 

4.33 

4QO 

.954090 

1.03 

.685934 

5.37 

5  QK 

.314066 

7 

54 
55 
56 
57 

.640284 
.640544 
.640804 
.641064 

.00 
4.33 
4.33 
4.33 

.954029 
.953968 
.953906 
.953845 

1.02 
1.02 
1.03 
1.02 

.686255 
.686577 
.686898 
.687219 

.uO 

5.37 
5.35 
5.35 

.313745 
.313423 
.313102 
.312781 

6 
5 

4 
3 

58 
59 
GO 

.641324 
.641583 
9.641842 

4.33 
4.32 
4.32 

.953783 
.953722 
9.953660 

1.03 
1.02 
1.03 

.687540 
.687861 
9.688182 

5.35 
5.35 
5.35 

.312460 
.312139 
10.311818 

2 
1 
0 

'  |  Cosine. 

D.  1'. 

Sine. 

D.  r.  1 

|  Cotang. 

D.  1".   Tang. 

' 

115° 


TABLE   XII.      LOGARITHMIC    SINES, 


153* 


' 

Sine. 

D.  1". 

Cosine. 

D.I". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

9.641842 
.642101 

4.32 

439 

9.953660 
.953599 

1.02 

1  03 

9.688182 
.688502 

5.33 

10.311818  60 
.311498  59 

2 
3 
4 

.642360 
j  .642618 
.642877 

.06 

4.30 
4.32 
4  30 

.953537 
.953475 
.953413 

1  .Uo 

1.03 
1.03 

.688823 
.689143 
.689463 

5^33 
5.33 

533 

.311177  58 
.31C857  57 
.310537  KB 

5 
6 

.643135 
.643393 

4^30 

.953352 
.953290 

l.'OB 

.689783 
i  .690103 

.CO 

5.33 

.310217 
.309897 

54 

7 

.643650 

A  3ft 

.953228 

1"ft3 

.690423 

5.33 

.309577 

53 

8 
9 
10 

.643908 
.644165 
.644423 

4^28 
4.30 
4.28 

.953166 
.953104 
.953042 

.Uo 

1.03 
1.03 
1.03 

.690742 
.691062 
.691381 

5.32 
5.33 
5.32 
5.32 

.309258 
.308938 
.308619 

52 
51 
50 

11 

9.644680 

A  97 

9.952980 

1  03 

9.691700 

539 

10.308300 

49 

12 
13 
14 
15 
16 
17 

.644936 
.645193 
.645450 
.645706 
.645962 
.646218 

4.*< 

4.28 
4.28 
4.27 
4.27 
4.27 

A  97 

.952918 
,952855 
.952793 
.952731 
.952669 
.952606 

I'M 

1.03 
1.03 
1.05 

.692019 
.692338 
.692656 
.692975 
.693293 
.693612 

.0/6 

5,32 
5.30 
5.32 
5.30 
5.32 

53ft 

.307981 
.307662 
.307344 
.307025 
.306707 
.306388 

48 
47 
46 
45 
44 
43 

18 
19 
20 

.646474 
.646729 
.646984 

<±.,C4 

4.25 
4.25 
4.27 

.952544 
.952481 
.952419 

l.'Oo 
1.03 
1.05 

.693930 
.694248 
.694566 

.oU 

5.30 
5.30 
5.28 

.306070 
.305752 
.305434 

42 
41 
40 

21 

9.647240 

9.952356 

1  03 

9.694883 

53ft 

10.305117 

39 

22 
23 

24 

.647494 
.647749 
648004 

4^25 
4.25 
4.5i3 

.952294 
.952231 
.952168 

J[  .  Uo 

1.05 
1.05 
1  03 

.695201 
.695518 
.695836 

.oU 

5.28 
5.30 

.304799 
.304482 
.304164 

83 

36 

25 

.648258 

.952106 

1  .Uo 

1  05 

.696153 

5.28 

.303847 

35 

26 
27 
28 
29 
30 

.648512 
.648766 
.649020 
.649274 
.649527 

4.'23 
4.23 
4.23 
4.22 
4.23 

.952043 
.951980 
.951917 
.951854 
.951791 

l.'OS 
1  05 
1.05 
1.05 
1.05 

.696470 
.696787 
.697103 
.697420 
.697736 

5.28 
5.28 
5.27 
5.28 
5.27 
5.28 

.303530 
.303213 
.302897 
.302580 
.302264 

34 
33 
32 
31 
30 

31 
32 
33 
34 
35 

9.649781 
.650034 
.650287 
.650539 
.650792 

4.22 
4.22 
4.20 
4.22 

4  Oft 

9.951728 
.951665 
.951602 
.051539 
.951476 

1.05 
1.05 
1.05 
1.05 
1  07 

9.698053 
.698369 
.698685 
.699001 
.699316 

5.27 
5.27 
5.27 
5.25 

10.301947 
.301631 
.301315 
.300999 
.300684 

29 
28 
27 
26 
25 

36 
37 
38 
39 

40 

.651044 
.651297 
.651549 
.651800 
.652052 

.t!\J 

4.22 

4.20 
4.18 
4.20 
4.20 

.951412 
.951349 
.951286 
.951222 
.951159 

1  .Vi 

1  05 
1  05 
1.07 
1  05 
l!05 

.699032 
.699947 
.700263 
.700578 
.700893 

5.27 
5.25 
5.27 
5.25 
5.25 
5.25 

.300368 
.300053 
.299737 
.299422 
.299107 

24 

23 

21 

20 

41 
42 

9.652304 
.652555 

4.18 

A  1ft 

9.951096 
.951032 

1.07 

1  07 

9.701208 
.701523 

5.25 

10.298792 
.298477 

19 
18 

43 
44 
45 

.652806 
.653057 
653308 

"*.  lo 

4.18 
4.18 

417 

.950968 
.950905 
.950841 

i!<* 

1.07 
1.05 

.701837 
.702152 
.702466 

5^25 
5.23 

.298163 
.297848 
.297534 

17 
16 
15 

46 
47 

48 

.653558 
.653808 
.654059 

.it 

4.17 

4.18 

417 

.950778 
.950714 
.950650 

1.07 
1-°Z 

.702781 
.703095 
.703409 

5.25 
5.23 
5.23 

.297219 
.296905 
.296591 

14 

13 
12 

49 
50 

.654309 
.654558 

.if 

4.15 
4.17 

.950586 
.950522 

l!07 
1.07 

.703722 
.704036 

5.22 
5.23 
5.23 

.296278 
.295964 

11 

10 

51 
52 
53 

9.654808 
.655058 
.655307 

4.17 
4.15 

4-JK 

9.950458 
.950394 
.950330 

1.07 
1-°l 

9.704350 
.704663 
.704976 

5.22 
5.22 

593 

10.295650 
.295337 
.295024 

9 

8 
7 

54 
55 
56 
57 
58 
59 
60 

.655556 
.655805 
.656054 
656302 
.656551 
.656799 
9.657047 

.  ID 

4.15 
4.15 
4.13 
4.15 
4.13 
4.13 

.950266 
.950202 
.950138 
.950074 
.950010 
.949945 
9.949881 

li07 
1.07 
1.07 
1.07 

1.08 
1.07 

.705290 
.705603 
.705916 
.706228 
.706541 
.706854 
9.707166 

.39 

5.22 
5.22 
5.20 
5.22 
5.22 
5.20 

.294710 
.294397 
.294084 
.293772 
.293459 
.293146 
10.292834 

6 
5 
4 
3 
2 
1 
0 

/  ^ 

Cosine. 

D.I". 

Sine.   D.  1".  ,  Cotaog. 

D.  1".  1 

Tang. 

' 

63' 


COSIKES,  TAKGEHTS, 


COTAHGEHTS. 


t 

Sine. 

D.r. 

Cosine. 

D.I". 

Tang. 

D.I". 

Cotang. 

' 

0 

9.657047 

9  949881 

Iflft 

9  707166 

10.292834 

60 

1 

2 
3 

4 
5 
6 

7 

.657295 
.657542 
.657790 
.658037 

.658284 
.658531 
.658778 

4^12 
4.13 
4  12 
4.12 
4.12 
4.12 
410 

.949816 
.949752 
.949688 
.949623 
.  949558 
.949494 
.949429 

.Uo 

1  07 
1.07 
1.08 
1.08 
1.07 
1.08 

IAft 

707478 
.7C7790 
708102 
.708414 
.708726 
709037 
.709349 

5  20 
5  20 
5.20 
5  20 
5  18 
5  20 
51ft 

.292522 
.292210 
.291898 
.291586 
.291274 
.290963 
.200651 

59 

58 
57 
56 
55 
54 
53 

8 
9 
10 

.659025 
.659271 
.659517 

1.5 

4.10 
4.10 
4.10 

.949364 
.949300 
.949235 

.Uo 

1.07 
1.08 
1.08 

.709600 
.709971 
.710282 

.  Jo 
5.18 
5.18 
5.18 

.290340 
.290029 
.289718 

52 
51 
50 

11 

9-659763 

9.949170 

9.710593 

51ft 

10.289407 

49 

12 

.660009 

4.10 

.949105 

1.08 

.710904 

.  10 

.289096 

48 

13 

.660255 

4  10 

.949040 

1.08 

.711215 

5.18 

f  17 

.288785 

47 

14 

.660501 

4  10 

.948975 

1  .08 

.711525 

O.ll 

.  .288475 

46 

15 

.660746 

4  08 

.948910 

1.08 

.711836 

5.18 

.288164 

45 

16 
17 

.660991 
.661236 

4.08 
4.08 

.948845 
.948780 

1.08 
1.08 

.712146 
.712456 

5.17 
5.17 

.287854 
.287544 

44 
43 

18 
19 
20 

.661481 
.661726 
.661970 

4.08 
4.08 
4.07 
4  07 

.948715 
.948650 
.948584 

1.08 
1.08 
1.10 
1  08 

.712706 
.713076 
.713386 

5.17 
5  17 
5.17 
5.17 

.287234 
.286924 
.286614 

42 
41 
40 

21 

9.662214 

4  no 

9.948519 

IAft 

9.713696 

5  15 

10.286304 

39 

22 
23 
24 
25 

.662459 
.662703 
.662946 
.663190 

.Uo 

4.07 
4.05 
4.07 

.948454 
.948388 
.948323 
.948257 

.Uo 

1.10 
1  08 
1.10 

.714005 
.714314 
.714624 
.714933 

5.15 
5.17 
5  15 

.285995 
.285686 
.285376 
.285067 

38 
37 
36 
35 

26 
27 
28 
29 
30 

.663433 
.663677 
.663920 
.664163 
.664406 

4.05 
4.07 
4.05 
4.05 
4  05 
4.03 

.948192 
.948126 
.948060 
.947995 
.947929 

1.08 
1  10 
1.10 
1  08 
1.10 
1.10 

.715242 
.715551 
.715860 
.716168 
.716477 

5.15 
5.15 
5.15 
5  13 
5.15 
5.13 

.284758 
.284449 
.284140 
.283832 
.283523 

34 
33 
32 
31 
30 

31 

32 

9.664648 
.664891 

4  05 

9.947863 
.947797 

1.10 

9.716785 
.717093 

5.13 

10.283215 

.282907 

29 

28 

33 
34 

.665133 
.665375 

4.03 
4.03 

.947731 

.947GG5 

1.10 
1.10 

.717401 
.717709 

5  13 
5.13 

.282599 
.282291 

27 
26 

35 
36 
37 
38 
39 
40 

.665617 
.665859 
.666100 
.666342 
.666583 
.666824 

4.03 
4.03 
4.02 
4.03 
4.02 
4.02 
4.02 

.947600 
.947533 
.947467 
.947401 
.947335 
.947269 

1.08 
1.12 
1.10 
1.10 
1.10 
1.10 
1.10 

.718017 
.718325 
.718633 
.718940 
.719248 
.719555 

5.13 
5.13 
5.13 
5.12 
5.13 
5.12 
5.12 

.281983 
.281675 
.281367 
.281060 
.280752 
.280445 

25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

9.667065 
.667305 
.667546 
.66T786 
.668027 
.668267 
.668506 
.668746 
.668986 
.669225 

4.00 
4.02 
4.00 
4.02 
4.00 
3.98 
4.00 
4.00 
3.98 
3.98 

9.947203 
.947136 
.947070 
.947004 
.946937 
.946871 
.946804 
.946738 
.946671 
.946604 

1.12 
1.10 
1.10 
1.12 
1.10 
1.12 
1.10 
.12 
.12 
.10 

9.719862 
.720109 
.720476 
.720783 
.721089 
.721396 
.721702 
.722009 
.722315 
.722621 

5.12 
5.12 
5.12 
5.10 
5.12 
5.10 
5.12 
5.10 
5.10 
5.10 

10.280138 
.279831 
.279524 
.279217 
.278911 
.278604 
.278298 
.277991 
.277685 
.277379 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 

9.669464 

3OQ 

9  946538 

9.722927 

10.277073 

9 

62 
53 
54 
55 

56 
57 
58 
59 

.669703 
.669942 
670181 
.670419 
.670658 
.670896 
.671134 
.671372 

yo 
3.98 
3.98 
3.97 
3.98 
3.97 
3.97 
3.97 

.946471 
.946404 
.946337 
.946270 
.9462C3 
.946136 
.946069 
.946002 

.12 
.12 
.12 
.12 
.12 
.12 
12 
!l2 

.723232 
.723538 
.723844 
.724149 
.724454 
.724760 
.725065 
.725370 

5.08 
5.10 
5.10 
5.08 
5.08 
5.10 
5.08 
5.08 

.27'67G8 
.276462 
.276156 
.275851 
.275546 
.275240 
.274935 
.274630 

8 
7 
6 
5 
4 
3 
2 
1 

60 

9.671609 

3.95 

9.945935 

.12 

9.725674 

5.07 

10.274326 

0 

• 

Cosine. 

D.  1*. 

Sine. 

D.  1*. 

i  Cotang. 

D.I". 

Tang. 

' 

117* 


TABLE    XII.       LOGARITHMIC   SIKES, 


151* 


' 

Sine. 

D.I'. 

Cosine. 

D.  r. 

Tang. 

D.r. 

Cotang. 

- 

0 

1 

2 

9.671609 
.671847 

.672084 

3.97 
3.95 

9.945935 
.945868 
.945800 

1.12 
1.13 

9.725674 
.725979 
.726284 

5.08 
5.08 

10.274326 
.274021 
.273716 

60 

59 

58 

3 
4 
5 

.672321 
.672558 
.672795 

3.95 
3.95 
3.95 

.945733 
.945666 
.945598 

1.12 
1.12 
1.13 

.726588 
.726892 
.727197 

5.07 
5.07 
5.05 

.273412 
.273108 
.272803 

57 
56 
55 

6 

7 

.673032 
.673268 

3.95 
3.93 

.945531 
.945464 

1.12 

.12 

.727501 

.727805 

5.07 
5.07 

.272499 
.272195 

54 
53 

8 
9 

.673505 
.673741 

3.95 
3.93 

.945396 
.945328 

.13 
.13 

.728109 

.728412 

5.07 
5.05 

.271891 
.271588 

52 
51 

10 

.673977 

3.93 

.945261 

.12 

.728716 

5.07 

.271284 

50 

3.93 

.13 

5.07 

11 

9.674213 

9.945193 

9.729020 

10.270980 

49 

12 
13 

.674448 
.674684 

3.92 
3.93 

.945125 
.945058 

'.  .13 
.12 

.729323 
.729626 

5.05 
5.05 

.270677 
.270374 

48 
47 

14 
15 
16 

.674919 
.675155 
.675390 

3.92 
3.93 
3.92 

.944990 
.944922 
.944854 

1.13 
1.13 
1.13 

.729929 
.730233 
.730535 

5.05 
5.07 
5.03 

.270071 
.269767 
.269465 

46 
45 
44 

17 

18 

.675624 
.675859 

3.90 
3.92 

.944786 
.944718 

1.13 
1.13 

.730838 
.731141 

5.05 
5.05 

.269162 
.268859 

43 

42 

19 

.676094 

3.92 

.944650 

1.13 

.731444 

5.05 

.268556 

41 

20 

.676328 

3.90 

.944582 

1  .13 

.731746 

5.03 

.268254 

40 

3.90 

1.13 

5.03 

21 

22 

9.676562 
.67'6796 

3.90 

9.944514 
.944446 

1.13 

9.732048 
.732351 

5.05 

10.267952 
.267649 

39 
38 

23 
24 
25 

.677030 
.677264 
.677498 

3.90 
3.90 
3.90 

.944377 
.944309 
.944241 

1.15 
1.13 
1.13 

.732653 
.732955 
.733257 

5.03 
5.03 
5.03 

.267347 
.267045 
.266743 

37 
36 
35 

26 

.677731 

3.88 

.944172 

1.15 

.733558 

5.02 

.266442 

34 

27 

28 

.677964 
.678197 

3.88 
3.88 

.944104 
.944036 

1.13 
1.13 

.733860 
.734162 

5.03 
5.03 

.266140 
.265838 

33 

32 

29 

.678430 

3.88 

.943967 

1.15 

.734463 

5  .02 

.265537 

31 

30 

.678663 

3.88 
3.87 

.943899 

1.13 
1.15 

.734764 

5.02 
5.03 

.265^66 

30 

31 
32 
33 

9.678895 
.679128 
.679360 

3.88 
3.87 

9.943830 
.943761 
.943693 

1.15 
1.13 

9.735066 
.735367 
.735668 

5.02 
5.02 

10.264934 
.264633 
-.264332 

29 

28 
27 

34 

.679592 

3.87 

.943624 

1.15 

.735969 

5.02 

.264031 

26 

35 
36 
37 
38 
39 

.679824 
.680056 
.680288 
.680519 
.680750 

3.87 
3.87 
3.87 
3.85 

3.85 

.943555 
.943486 
.943417 
.943348 
.943279 

1.15 
1.15 
1.15 
1.15 
1.15 

.736269 
.736570 
.736870 
.737171 
.737471 

5.00 
5.02 
5.00 
5.02 
5.00 

.263731 
.263430 
.263130 
.262829 
.262529 

25 
24 
23 
22 
21 

40 

.680982 

3.87 
3.85 

.943210 

1.15 
1.15 

.737771 

5.00 
5.00 

.262229 

20 

41 
42 
43 
44 
45 
46 
47 

9.681213 
.681443 
.681674 
.681905 
.682135 
.682365 
.682595 

3.83 
3.85 
3.85 
3.83 
3.83 
3.83 

9.943141 
.94':072 
.943003 
.942934 
.942864 
.942795 
.942726 

1.15 
1.15 
1.15 
1.17 
1.15 
1.15 

9.738071 
.738371 
.738671 
.738971 
.739271 
.739570 
.739870 

5.00 
5.00 
5.00 
5.00 
4.98 
5.00 

10.261929 
.261629 
.261329 
.261029 
.260729 
.260430 
.260130 

19 
18 
17 
16 
15 
14 
13 

48 

.682825 

3.83 

.942656 

1.17 

.740169 

4.98 

4  Oft 

.259831 

12 

49 
50 

.683055 
.683284 

3.83 
3.82 
3.83 

.942587 
.942517 

1.15 
1.17 
1.15 

.740468 
.740767 

.Uo 
4.98 
4.98 

.259532 
.259233 

11 
10 

51 

62 

9.683514 
.683743 

3.82 

9.942448 
.942378 

1.17 

9.741066 
.741365 

4.98 

10.258934 
.258635 

9 

8 

53 

.683972 

3.82 

3QO 

.942308 

1.17 

1-tK 

.741664 

4.98 

A  Off 

.258336 

7 

54 
55 

.684201 
.684430 

.Ofi 

3.82 

.942239 
.942169 

.  1O 

1.17 

.741962 
.742261 

<t.  y< 
4.98 

.258038 
.257739 

6 
5 

56 

.684658 

3.80 

.942099 

1.17 

.742559 

4.  97 

.257441 

4 

57 
58 
59 
60 

.684887 
.685115 
.685343 
9.685571 

3.82 
3.80 
3.80 
3.80 

.942029 
.941959 
.941889 
9.941819 

1.17 
1.17 
1.17 
1.17 

.742858 
.743156 
.743454 
9.743752 

4.98 

4^97 
4.97 

.257142 
.256844 
.256546 
10.256248 

3 
2 

0 

' 

Cosine. 

D.  r. 

Sine. 

D.I". 

Cotang. 

D.  r. 

Tang. 

' 

61" 


COSINES,   TANGENTS,   AND    COTANGENTS. 


/ 

Sine. 

D.l'. 

Cosine. 

D.r. 

Tang. 

D.  1'. 

Cotang. 

' 

0 

1 

2 

9.685571 
.685799 
.686027 

3.80 

3.80 
370 

9.941819 
.941749 
.941679 

1.17 
1.17 

1  17 

9.743752 
.744050 

.744348 

4.97 
4.97 

A  QK 

10.256248 
.255950 
.255652 

60^ 
59 

58 

3 

.686254 

.  <o 

.941609 

JL.J.I 

1  17 

.744645 

•i.yo 
4  97 

.255355 

57 

4 
5 
6 
7 
8 
9 
10 

.686482 
.686709 
.686936 
.687163 
.687389 
.687616 
.687843 

3>8 
3.78 
3.78 
3.77 
3.78 
3.78 
3.77 

.941539 
.941469 
.941398 
.941328 
.941258 
.941187 
.941117 

l.lf 

1.17 
1.18 
1.17 
1.17 
1.18 
1.17 
1.18 

.744943 
.745240 
.745538 

.74^835 
.74ol32 
.746429 
.746726 

4.  95 
4.97 
4.95 
4.95 
4.95 
4.95 
4.95 

.255057 
.254760 
.254462 
.254165 
.253868 
.253571 
.253274 

56 
55 
54 
53 

52 
51 
50 

11 
12 
13 
14 

9.  688069 
.688295 
.688521 
.688747 

3.77 
3.77 
3.77 

37K 

9.941046 
.940975 
.940905 
.940834 

1.18 
1.17 
1.18 
11ft 

9.747023 
.747319 
.747616 
.747913 

4.93 
4.95 
4.95 

10.252977 
.252681 
.252384 
.252087 

49 
48 
47 
46 

15 
16 
17 
18 
19 
20 

.688972 
.689198 
.689423 
.689648 
.689873 
.690098 

.  (O 

3.77 
3.75 
3.75 
3.75 
3.75 
3.75 

.940763 
.940693 
.940622 
.940551 
.940480 
.940409 

.  lo 

1.17 
1.18 
1.18 
1.18 
1.18 
1.18 

.748209 
.748505 
.748801 
.749097 
.749393 
.749689 

4^93 
4.93 
4.93 
4.93 
4.93 
4.93 

.251791 
.251495 
.251199 
.250903 
.250607 
.250311 

45 
44 
43 
42 

41 
40 

21 

9.690323 

9.940338 

9.749985 

4QQ 

10.250015 

39 

22 
23 

.690548 
.690772 

3>3 

370 

.940267 
.940196 

1.18 
1.18 

.750281 
.750576 

.yo 
4.92 

4QQ 

.249719 
.249424 

38 
37 

24 
25 

.690996 
.691220 

.  1  0 

3.73 
370 

.940125 
.940054 

1.18 
1.18 

.750872 
.751167 

.yo 
4.92 
400 

.249128 
.248833 

36 
35 

26 
27 
28 
29 
30 

.691444 
.691668 
.691892 
.692115 
.692339 

.  Id 

3.73 
3.73 
3.72 
3.73 
3.72 

.939982 
.939911 
.939840 
.939768 
.939697 

l!l8 
1.18 
1.20 
1.18 
1.20 

.751462 
.751757 
.752052 
.752347 
.752642 

.KB 

4.92 
4.92 
4.92 
4.92 
4.92 

.248538 
.248243 
.247948 
.247653 
.247358 

34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.692562 
.692785 
.693008 
.693231 
.693453 
.693676 
.693898 
.694120 
.694342 
.694564 

3.72 
3.72 
3.72 
3.70 
3.72 
3.70 
3.70 
3.70 
3.70 
3.70 

9.939625 
.939554 
.939482 
.939410 
.939339 
.939267 
.939195 
.939123 
.939052 
.938980 

1.18 
1.20 
1  20 
1.18 
1.20 
1.20 
1.20 
1.18 
1.20 
1.20 

9.752937 
.753231 
.753526 
.753820 
.754115 
.754409 
.754703 
.754997 
.755291 
.755585 

4.90 

4.92 
4.90 
4.92 
4.90 
4.90 
4.90 
4.90 
4.90 
4.88 

10.247063 
.246769 
.246474 
.246180 
.245885 
.245591 
.245297 
.245003 
.244709 
.244415 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 

9.694786 
.695007 

3.68 

9.938908 
.938836 

1.20 

9.755878 
.756172 

4.90 

10.244122 

.243828 

19 
18 

43 

.695229 

3  CO 

.938763 

1.22 
Ion 

.756465 

4.88 

.243535 

17 

44 
45 

.695450 
.695671 

.Do 

3.68 

3  CO 

.938691 
.938619 

.  JiU 

1.20 

.756759 
.757052 

4.90 

4.88 

.243241 
.242948 

16 
15 

46 
47 

48 

.695892 
.696113 
.696334 

.  Do 

3.68 
3.68 
3  67 

.938547 
.938475 
.938402 

l!20 
1.22 

.757345 
.757638 
.757931 

4.88 
4.88 
4.88 

4QQ 

.242655 
.242362 
.242069 

14 
13 
12 

49 

.696554 

3OQ 

.938330 

i  on 

.758224 

.OO 

4  GO. 

.241776 

11 

50 

.696775 

.  Do 

3.67 

.938258 

lisa 

.758517 

.00 

4.88 

.241483 

10 

51 

52 
53 

9.696995 
.697215 
.697435 

3.67 
3.67 

3CK 

9.938185 
.938113 
.938040 

1.20 
1.22 

9.758810 
.759102 
.759395 

4.87 

4.88 

10.241190 

.240898 
.240605 

9 

8 
7 

54 
55 
56 
57 
58 
59 

.697654 
.697874 
.698094 
.698313 
.698532 
.698751 

.DO 

3.67 
3.67 
3.65 
3.65 
3.65 

3CK 

.937967 
.937895 
.937822 
.937749 
.937676 
.937604 

1.22 
1.20 
1.22 
1.22 
1.22 
1.20 

.759687 
.759979 
.760272 
.760564 
.760856 
.761148 

4.87 
4.87 
4.88 
4.87 
4.87 
4.87 

4  OK 

.240313 
.240021 
.239728 
.239436 
.239144 
.238852 

6 
5 
4 
3 
2 
1 

60 

9.698970 

.DO 

9.937531 

1.22 

9.761439 

.oO 

10.238561 

0 

' 

Cosine. 

D.  r. 

Sine. 

D.r. 

Cotang. 

D.r. 

Tang.  1 

' 

9.0.7 


TABLE   XII.      LOGARITHMIC   SINES, 


' 

Sine. 

D.  1". 

Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

2 
3 
4 
5 
6 

9.698970 
.699189 
.699407 
.699626 
.699844 
.700062 
.700280 

3.65 
3.63 
3.65 
3.63 
3.63 
3.63 

9.937531 
.937458 
.937385 
.937312 
.937238 
.937165 
.937092 

1.22 
1.22 
1.22 
1.23 
1.22 
1.22 

9.761439 
.761731 
.762023 
.762314 
.762606 
.762897 
.763188 

4.87 
4.87 
4.85 
4.87 
4.85 
4.85 

10.238561 
.288269 
.237977 
.237686 
.237394 
.237103 
.236812 

60 
59 
58 
57 
56 
55 
54 

7 
8 
9 
10 

.700498 
.700716 
.700933 
.701151 

3.63 
3.63 
3.62 
3.63 
3.62 

.937019 
.936946 
.936872 
.936799 

1.22 
1.22 
1.23 
1.22 
1.23 

.763479 
.763770 
.764061 
.764352 

4.85 
4.85 
4.85 
4.85 
4.85 

.236521 
.236230 
.235939 
.235648 

53 
52 
51 
50 

11 
12 
13 
14 
15 
16 

9.701368 
.701585 
.701802 
.702019 
.702236 
.702452 

3.62 
3.62 
3.62 
3.62 
3.60 
3  62 

9.936725 
.936652 
.936578 
.936505 
.936431 
.936357 

1.22 
1.23 
1.22 
1.23 
1.23 
1  22 

9.764643 
.764933 
.765224 
.765514 
.765805 
.766095 

4.83 
4.85 
4.83 
4.85 
4.83 

4  DO 

10.235357 
.235067 
.234776 
.234486 
.234195 
.233905 

49 

48 
47 
46 
45 
44 

17 

.702669 

3  60 

.936284 

.766385 

OO 

.233615 

43 

18 
19 
20 

.702885 
.703101 
.703317 

s!eo 

3.60 
3.60 

.936210 
.936136 
.936062 

l!23 
1.23 
1.23 

.766675 
.766965 
.767255 

4.83 
4.83 
4.83 
4.83 

.233325 
.233035 
.232745 

42 

41 
40 

21 
22 

9.703533 
.703749 

8.60 
3  58 

9.935988 
.935914 

1.23 

9.767545 

.767834 

4.82 

10.232455 
.232166 

39 

38 

23 

.703964 

3  58 

.935840 

1  OO 

.768124 

4.83 

400 

.231876 

87 

24 

.704179 

3  fin 

.935766 

1  .40 

.768414 

.OO 

.231586 

36 

25 
26 

27 

.704395 
.704610 
.704825 

.  DU 

3.58 
3.58 

3KO 

.935692 
.935618 
.935543 

1^23 
1.25 

.768703 
.768992 
.769281 

4.82 
4.82 
4.82 

.231297 
.231008 
.230719 

35 
34 
33 

28 
29 
30 

.705040 
.705254 
.705469 

.OO 

3.57 
3.58 
3.57 

.935469 
.935395 
.935320 

1  .23 

l!25 
1.23 

.769571 
.769860 
.770148 

4.83 
4.82 
4.80 
4.82 

.230429 
.230140 

.229852 

32 
31 
30 

31 

9.705683 

3  58 

9.935246 

1OK 

9.770437 

409 

10.229563 

29 

32 

.705898 

.935171 

.  4O 

193 

.770726 

.04 

.229274 

28 

33 

.706112 

3  57 

.935097 

.40 

.771015 

4.82 

.228985 

27 

34 

.706326 

3  55 

.935022 

1  93 

.771303 

4.80 
409 

.228697 

26 

35 
36 
37 
38 

.706539 
.706753 
.706967 
.707180 

3.57 
3.57 
3.55 
3  '55 

.934948 
.934873 
.934798 
.934723 

1  .40 

1.25 
1.25 
1.25 
1  93 

.771592 
.771880 
.772168 
.772457 

.04 

4.80 
4.80 

4.82 

4QA 

.228408 
.228120 
.227832 
.227543 

25 
24 
23 
22 

39 

.707393 

3KK 

.934649 

1  ,4O 

19K 

.772745 

.OU 
4  DA 

.227255 

21 

40 

.707606 

.00 
3.55 

.934574 

./GO 

1.25 

.773033 

.oU 

4.80 

.226967 

20 

41 
42 
43 

9.707819 
.708032 
.708245 

3.55 
3.55 

3KK 

9.934499 
.934424 
.934349 

1.25 
1.25 

19K 

9.773321 

.773608 
.773896 

4.78 
4.80 

10.226679 
.226392 
.226104 

19 
18 
17 

44 

.708458 

.OO 

3  53 

.934274 

.  /cO 

.774184 

470 

.225816 

16 

45 

.708670 

3KO 

.934199 

197 

.774471 

.  to 

.225529 

15 

46 
47 

48 

.708882 
.709094 
.709306 

.OO 

3.53 
3.53 

3KO 

.934123 
.934048 
.933973 

.4i 

1.25 
1.25 

.774759 
.775046 
.775333 

4.80 
4.78 
4.78 

4  DA 

.225241 
.224954 
.224667 

14 
13 
12 

49 
50 

.709518 
.709730 

.OO 

3.53 
3.52 

.933898 
I  .933822 

l!27 
1.25 

.775621 
.775908 

.ou 
4.78 
4.78 

.224379 
.224092 

11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.709941 
.710153 
.710364 
.710575 
.710786 
.710997 
.711208 
.711419 
.711629 
9.711839 

3.53 
3.52 
3.52 
3.52 
3.52 
3.52 
3.52 
3.50 
3.50 

9.933747 
.933671 
.933596 
.933520 
.933445 
.933369 
933293 
.933217 
.933141 
9.933066 

1.27 
1  25 
1.27 
1.25 
1.27 
1.27 
1.27 
1.27 
1.25 

9.776195 

.776482 
.776768 
.777055 
.777342 
.777628 
.777915 
.778201 
.778488 
9.778774 

4.78 

4.77 
4.78 
4.78 
4.77 
4.78 
4.77 
4.78 
4.77 

10.223805 
.223518 
.223232 
.222945 
.222658 
.222373 
.222085 
.221799 
.221512 
10.221226 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

1 

Cosine. 

D.  1'. 

Sine. 

D.I'. 

Cotang. 

D.I',  i 

Tang. 

' 

31° 


COSINES,   TANGENTS,    AND   COTANGENTS. 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.I". 

Cotang. 

' 

0 

1 

2 

9.711839 
.712050 
.712260 

3.52 
3.50 

3AO 

9.933066 
.932990 
.932914 

1.27 
1.27 

197 

9.778774 
.779060 
.779346 

4.77 
4.77 

477 

10.221226 
.220940 
.220654 

60 

59 
58 

3 
4 
5 

.712469 
.712679 
.712889 

.<±o 
3.50 
3.50 

.932838 
.932762 
.932685 

.lit 

1.27 
1.28 

1  97 

.779632 
.779918 

.780203 

.  I  i 

4.77 
4.75 

.220368 
.220082 
.219797 

57 
56 
55 

6 

7 

.713098 
.713308 

3^50 

.932609 
.932533 

l.JW 

1.27 

197 

.780489 
.780775 

4.77 
4  77 

.219511 
.219225 

54 
53 

8 

,  9 
10 

.713517 
.713726 
.713935 

3!48 
3.48 
3.48 

.932457 
.932380 
.932304 

»*f 

1.28 
1.27 
1.27 

.781060 
.781346 
.781631 

4.75 
4.77 
4.75 
4.75 

.218940 
.218654 
.218369 

52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 

9.714144 
.714352 
.714561 
.714769 
.714978 
.715186 
.715394 
.715602 

3.47 
3.48 
3.47 
3.48 
3.47 
3.47 
3.47 

q  AK. 

9.932228 
.932151 
.932075 
.931998 
.931921 
.931845 
.931768 
.931691 

1.28 
1.27 
1.28 
1.28 
1.27 
1.28 
1.28 

1  9ft 

9.781916 
.782201 
.782486 
.782771 
.783056 
.783341 
.783626 
.783910 

4.75 
4.75 
4.75 
4.75 
4.75 
4.75 
4.73 

4tTK 

10.218084 
.217799 
.217514 
.217229 
.216944 
.216659 
.216374 
.216090 

49 
48 
47 
46 
45 
44 
43 
42 

19 
20 

.715809 
.716017 

o.'lO 

3.47 
3.45 

.931014 
.931537 

1  .  *O 

1.28 
1.28 

.784195 
.784479 

:.  <O 

4.73 
4.75 

.215805 
.215521 

41 
40 

21 
22 
23 
24 

9.716224 
.716432 
.716639 
.716846 

3.47 
3.45 
3.45 

34K 

9.931460 
.931383 
.931306 
.931229 

1.28 

1.28 
1.28 

1  9ft 

9.784764 
.785048 
.785332 
.785616 

4.73 
4.73 
4.73 

10.215236 
.214952 
.214668 
.214384 

39 
38 
37 
36 

25 
26 
27 

28 

.717053 
.717259 
.717466 
.717673 

.'.cO 

3.43 
3.45 
3.45 

.931152 
.931075 
.930998 
.930921 

1  ./VO 

1.28 
1.28 
1.28 

IQA 

.785900 
.786184 
.786468 
.786752 

4.73 
4.73 
4.73 

4.73 

47°. 

.214100 
.213816 
.213532 
.213248 

35 
34 
33 
32 

29 

.717879 

q  AQ 

.930843 

.OU 

1  9ft 

.787036 

.  (6 

.212964 

31 

30 

.718085 

O.4O 

3.43 

.930766 

l.-co 
1.30 

.787319 

4.72 
4.73 

.212681 

30 

31 
32 
33 
34 
35 

9.718291 
.718497 
.718703 
.718909 
.719114 

3.43 
3.43 
3.43 
3.42 

34P. 

9.930688 
.930611 
.930533 
.930456 
.930378 

1.28 
1.30 
1.28 
1.30 

9.787603 
.787'886 
.788170 
.788453 
.788736 

4.72 
4.73 
4.72 

4-Z2 

10.212397 
.212114 
.211830 
.211547 
.211264 

29 
28 
27 
26 
25 

36 
37 
38 
39 

;719320 
.719525 
.719730 
.719935 

.40 

3.42 
3.42 
3.42 

3  An 

.930300 
.930223 
.930145 
.930067 

l!28 
1  30 
1.30 

.789019 
.789302 
.789585 
.789868 

4.72 
4.72 

4.72 

.210981 
.210698 
.210415 
.210132 

24 
23 
22 
21 

40 

.720140 

."*/« 

3.42 

.929989 

1^30 

.790151 

4!  72 

.209849 

20 

41 

9.720345 

rt 

9.929911 

°/> 

9.790434 

10.209566 

19 

42 

.720549 

?*2? 

.929833 

1"  °.ft 

.790716 

4.  (0 

.209284 

18 

43 

.720754 

3.42 

.929755 

.ou 

1-qrj 

.790999 

4.72 

.209001 

17 

44 
45 
46 
47 

.720958 
.721162 
.721366 
.721570 

3.40 
3.40 
3.40 
3.40 

.929677 
.929599 
.929521 
.929442 

.ou 
1.30 
1.30 
1.32 

.791281 
.791563 
.791846 
.792128 

4.70 
4.70 
4.72 

4'Z° 

.208719 
.208437 
.208154 
.207872 

16 
15 
14 
13 

48 
49 

.721774 
.721978 

3.40 
3.40 

.929364 

.929286 

liao 

.792410 
.792692 

4  '.70 

.207590 
.207308 

12 
11 

50 

.722181 

3.38 
3.40 

.929207 

l!30 

.792974 

4.70 
4.70 

.207026 

10 

51 

9.722385 

9.929129 

100 

9.793256 

4  70 

10.206744 

9 

52 

.722588 

0.08 

.929050 

.«• 

Inn 

.793538  - 

.206462 

8 

53 

.722791 

8.38 

.928972 

.OU 

1  32 

.793819 

4.68 
470 

.206181 

7 

54 

.722994 

n  net 

.928893 

j  "qft 

.794101 

.  <u 

.205899 

6 

55 

.723197 

3.OO 
3qo 

.928815 

l.oU 

.794383 

4.70 

4AQ. 

.205617 

5 

56 

.723400 

.OO 

.928736 

1  *qo 

.794664 

.08 

.205336 

4 

57 

58 

.723603 
.723805 

3.38 
3.37 

.928657 
.928578 

1.0/6 

1.32 

1  °.9 

.794946 
.795227 

4.70 
4.68 

4  no 

.205054 
.204773 

3 
2 

59 
60 

.724007 
9.724210 

3.37 
3.38 

.928499 
9.928420 

1  .0* 

1.32 

.795508 
9.795789 

.DO 

4.68 

.204492 
10.204211 

1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  r. 

Cotang. 

D.  r. 

Tang. 

' 

TABLE   XII.      LOGARITHMIC    SIXES, 


' 

Sine. 

D.  1*. 

Cosine. 

D.  r. 

Tang. 

D.I". 

Cotang. 

i 

0 

1 

9.724210 
.724412 

3.37 

3q  ry 

9.928420 
.928342 

1.30 

9.795789 
.796070 

4.68 
4Rft 

10.204211 
.203930 

60 
59 

2 

.724614 

.6i 

.928263 

1.32 

.796351 

.  Do 

4fift 

.203649 

58 

3 

.724816 

3qK 

.928183 

1  QO 

.796632 

.Do 
4  Aft 

.203368 

57 

4 

.725017 

.OO 

3q>j» 

.928104 

1  QO 

.796913 

.Do 

4  CO 

.203087 

56 

5 
6 

.725219 
.725420 

.61 

3.35 

3q7 

.928025 
.927946 

l!32 

.797194 

.797474 

.Do 

4.67 

4  CO 

.202806 
.202526 

55 
54 

7 

.725622 

.6( 
3qc 

.927867 

Iqq 

.797755 

.  Do 
400 

.202245 

53 

8 
9 
10 

.725823 
.726024 
.726225 

.OO 

3.35 
3.35 
3.35 

.927787 
.927708 
.927623 

.00 

1.32 
1.32 
1.33 

.798036 
.798316 
.798596 

.DO 

4.67 
4.67 
4.68 

.201964 
.201684 
.201404 

52 
51 
50 

11 

9.726426 

3qq 

9.927549 

1  9.Q 

9.798877 

10.201123 

49 

12 
13 
14 
15 

.726626 
.726827 
.727027 
.727228 

.00 
3.35 
3.33 
3.35 

3qq 

.927470 
.927390 
.927310 
.927231 

1^33 
1.33 
1.32 

Iqq 

.799157 
.799437 
.799717 
.799997 

4.o< 
4.67 
4.67 
4.67 

.200843 
.200563 
.200283 
.200003 

48 
47 
46 
45 

16 

.727428 

.00 
3qq 

.927151 

.00 
Iqq 

.800277 

A  em 

.199723 

44 

17 
18 

.727628 

.727828 

.  Oo 

3.33 

3q9 

.927071 
.926991 

.OO 

1.33 

Iqq 

.800557 
.800836 

4^65 

.199443 
.199164 

43 
42 

19 

.728027 

.64 

.926911 

.00 

Iqq 

.801116 

A'  f»n 

.198884 

41 

20 

.728227 

3.33 
3.33 

.926831 

.00 

1.33  . 

.801396 

4.o7 
4.65 

.198604 

40 

21 

9.728427 

3  on 

9.926751 

Iqq 

9.801675 

4C.7 

10.198325 

39 

22 
23 

.728626 
.728825 

.<T4 
3.32 

3qO 

.926671 
.926591 

.OO 

1.33 

Iqq 

.801955 
.802234 

.D< 

4.65 

.198045 
.197766 

38 
37 

24 

.729024 

.64 

3qO 

.926511 

.66 

.802513 

4.65 

.197487 

36 

25 

26 

.729223 
.729422 

.64 

3.32 

3q9 

.926431 
.926351 

1.'33 

.802792 
.803072 

4^67 

4  OK 

.197208 
.196928 

35 
34 

27 
28 
29 
30 

.729621 
.729820 
.730018 
.730217 

,O4 

3.32 
3.30 
3.32 
3.30 

.926270 
.926190 
.926110 
.926029 

l!33 
1.33 
1.35 
1.33 

.803351 
.803630 
.803909 
.804187 

.DO 

4.65 
4.65 
4.63 
4.65 

.196649 
.196370 
.196091 
.195813 

33 

32 
31 
30 

31 

32 

9  730415 
.730613 

3.30 

3qfl 

9.925949 

.925868 

1.35 

Iqq 

9.804466 
.804745 

4.65 

10.195534 
.195255 

29 

28 

33 
34 

.730811 
.731009 

.OU 

3.30 

3  Oft 

.925788 
.925707 

.OO 

1.35 

.805023 
.805302 

4.63 
4  65 

.194977 
.194698 

27 
26 

35 
36 

.731206 
.731404 

,4o 
3.30 

3qfi 

.925626 
.925545 

1.35 
1.35 

Iqq 

.805580 
.805859 

4.63 
4.65 

.194420 
.194141 

25 
24 

37 
38 
39 
40 

.731602 
.731799 
.731996 
.732193 

.oU 

3.28 
3.28 
3.28 
3.28 

.925405 
.925384 
.925303 
.925222 

.Oo 

1.35 
1.35 
1.35 
1.35 

.806137 
.806415 
.806693 
.8Qg971 

4.63 
4.63 
4.63 
4.63 

.193863 
.193585 
.193307 
.193029 

23 
22 
21 
20 

41 

9.732390 

9.925141 

tqt» 

9.807249 

4  on 

10.192751 

19 

42 
43 
44 

.732587 
.732784 
.732980 

3^28 
3.27 

39ft 

.925060 
.924979 
.924897 

.oO 

1.35 
1-37 

.807527 
.807805 
.808083 

.Do 

4.63 
4.63 

.192473 
.192195 
.191917 

18 
17 
16 

45 

46 
47 
48 
49 
50 

.733177 
.733373 
.733569 
.733765 
.733961 
.734157 

.40 

3.27 
3.27 
3.27 
3.27 
3.27 
3.27 

.924816 
.924735 
.924654 
.924572 
.924491 
.924409 

1  .35 
1.35 
1.35 
1.37 
1.35 
1.37 
1.35 

.808361 
.808638 
.808916 
.809193 
.809471 
.809748 

4^62 
4.63 
4.62 
4.63 
4.62 
4.62 

.191639 
.191362 
.191084 
.190807 
.190529 
.190252 

15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.734353 
.734549 
.734744 
.734939 
.735135 
.735330 
.735525 
.735719 
.735914 
9.736109 

3.27 
3.25 
3.25 
3.27 
3.25 
3.25 
3.23 
3.25 
3.25 

9.924328 
.924246 
.924164 
.924083 
.924001 
.923919 
.923837 
.923755 
.923673 
9.923591 

1.37 
1.37 
1.35 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 

9.810025 
.810302 
.810580 
.810857 
.811134 
.811410 
.811687 
.811964 
.812241 
9.812517 

4.62 
4.63 
4.62 
4.62 
4.60 
4.62 
4.62 
4.62 
4.60 

10.189975 
.189698 
.189420 
.189143 
.188866 
.188590 
.188313 
.188036 
.187759 
10.187483 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  1". 

Sine. 

D.  1'. 

Cotang. 

D.  1'. 

Tang. 

' 

$7" 


33- 


COSINES,   TAKGEKTS,  AXD 


' 

Sine. 

D.  1". 

Cosine. 

D.  1'. 

Tang. 

D.I'. 

Cotang. 

' 

0 

9.736109 

9.923591 

9.812517 

10.187483 

60 

1 

2 
3 

4 
5 

.736303 
.736498 
.736692 
.736886 
.737080 

3.23 
3.25 
3.23 
3.23 
3.23 

.923509 
.923427 
.923345 
.923263 
.923181 

1.37 
1.37 
1.37 
1.37 

1QQ 

.812794 
.813070 
.813347 
.813623 
.813899 

4.62 
4.60 
4.62 
4.60 
4.60 

4A9 

.187206 
.186930 
.186653 
.186377 
.186101 

59 
58 
57 
56 
55 

6 

7 

.737274 
.737467 

3^22 
300 

.923098 
.923016 

.OO 

1.37 

1QQ 

.814176 
.814452 

.COS 

4.60 

4  Art 

.185824 
.185548 

54 
53 

•  8 
9 

.737661 

.737855 

,i6O 

3.23 

.922933 

.922851 

.OO 

1.37 

1OQ 

.814728 
.815004 

.DU 

4.60 

.185272 
.184996 

52 
51 

10 

.738048 

3.22 
3.22 

.922768 

.OO 

1.37 

.815280 

4.60 
4.58 

.184720 

50 

11 

9.738241 

9.922686 

InQ 

9.815555 

4  Art 

10.184445 

49 

12 
13 

.738434 
.738627 

3^22 

.922603 
.922520 

.OO 

1.38 

.815831 
.816107 

.DU 

4.60 

.184169 
.183893 

48 
47 

14 

.738820 

3.22 

.922438 

1  .37 

.816382 

4.58 

.183618 

46 

15 

.739013 

3.22 

.922355 

1  .38 

.816658 

4.60 

.183342 

45 

16 

.739206 

3.22 

.922272 

1  .38 

1   00 

.816933 

4.58 

.183067 

44 

17 

.739398 

3.20 

.922189 

1  .00 
10Q 

.817209 

4.60 

4  to 

.182791 

43 

18 
19 
20 

.739590 
.739783 
.739975 

3  22 
3.20 
3.20 

.922106 
.922023 
.921940 

OO 

1.38 
1.38 
1.38 

.817484 
.817759 
.818035 

.00 
4.58 
4.60 
4.58 

.182516 
.182241 
.181965 

42 
41 
40 

21 

9.740167 

3  20 

9.921857 

9.818310 

4  to 

10.181690 

39 

22 

.740359 

.  921774 

00 

.818585 

.00 

.181415 

38 

23 

.740550 

o  O0 

.921091 

1.08 

.818860 

4.58 

4fcO 

.181140 

37 

24 

25 

.740742 
.740934 

3^20 

.921607 
.921524 

lisa 

.819135 
.819410 

.00 
4.58 

.180865 
.180590 

36 
35 

26 

27 

.741125 
.741316 

3.18 
3.18 

.921441 
.921357 

1.38 
1.40 

.819684 
.819959 

4.57 
4.58 

.180316 
.180041 

34 
33 

28 
29 

.741508 
.741699 

3.20 
3.18 

31  7 

.921274 
.921190 

1.38 
1.40 

1OO 

.820234 
.820508 

4.58 
4.57 

.179766 
.179492 

32 
31 

30 

.741889 

.It 

3.18 

.921107 

.OO 

1.40 

.820783 

4.58 
4.57 

.179217 

30 

31 
32 

9.742080 
.742271 

3.18 

9.921023 

.920939 

1.40 

9.821057 
.821332 

4.58 

10.178943 
.178668 

29 
28 

33 

.742462 

3.18 

31  7 

.920856 

1.38 

.821606 

4.57 

4tiy 

.178394 

27 

34 
35 

36 
37 
38 
39 
40 

.742652 
.742842 
.743033 
.743223 
.743413 
.743602 
.743792 

.  -1  ( 

3.17 
3.18 
3.17 
3.17 
3.15 
3.17 
3.17 

.920772 
.920688 
.920604 
.920520 
.920436 
.920352 
.920268 

1^40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 

.821880 
.822154 
.822429 
.8227'03 
.822977 
.823251 
.823524 

.Ol 

4.57 
4.58 
4.57 
4.57 
4.57 
4.55 
4.57 

.178120 
.177846 
.177571 
.177^97 
.177023 
.176749 
.176476 

26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

9.743982 
.744171 
.744361 
.744550 
.744739 
.744928 
.745117 
.745306 
.745494 
.745683 

3.15 
3.17 
3.15 
3.15 
3.15 
3.15 
3.15 
3.13 
3.15 
3.13 

9.920184 
.920099 
.920015 
.919931 
.919846 
.919762 
.919677 
.919593 
.919508 
.919424 

1.42 
1.40 
1.40 
1.42 
1.40 
1.42 
1.40 
1.42 
1.40 
1.42 

9.823798 
.824072 
.824345 
.824619 
.824893 
.825166 
.825439 
.825713 
.825986 
.826259 

4.57 
4.55 
4.57 
4.57 
4.55 
4.55 
4.57 
4.55 
4.55 
4.55 

10.176202 
.175928 
.175655 
.175381 
.175107 
.174834 
.174561 
.174287 
.174014 
.173741 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.745871 
.746060 
.746248 
.746436 
.746624 
.746812 
.746999 
.747187 
.747374 
9.747562 

3.15 
3.13 
3.13 
3.13 
3.13 
3.12 
3.13 
3.12 
3.13 

9.919339 
.919254 
.919169 
.919085 
.919000 
.918915 
.918830 
.918745 
.918659 
9.918574 

1.42 
1.42 
1.40 
1.42 
1.42 
1.42 
1.42 
1.43 
1.42 

9.826532 
.826805 
.827078 
.827351 
.827624 
.827897 
.828170 
.828442 
.828715 
9.828987 

4.55 
4.55 
4.55 
4.55 
4.55 
4.55 
4.53 
4.55 
4.53 

10.173468 
.173195 
.172922 
.172649 
.172376 
.172103 
.171830 
.171558 
.171285 
10.171013 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  1". 

Sine. 

D.  1'. 

Cotang. 

D.  r. 

Tang. 

' 

84- 


TABLE  XII.      LOGARITHMIC    SINES, 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.I". 

Cotang. 

' 

0 

1 

2 

9.747562 
.747749 
.747936 

3.12 
3.12 

9.918574 
.918489 
.918404 

1.42 

1.42 

9.828987 
.829260 
.829532 

4.55 
4.53 

10.171013 
.170740 
.170468 

60 
59 

58 

3 

4 
5 
6 

7 

.748123 
.748310 
.748497 

.748683 
.748870 

3.12 
3.12 
3.12 
3.10 
3.12 

.918318 
.918233 
.918147 
.918062 
.917976 

1.43 
1.42 
1.43 
1.42 
1.43 

1A9 

.829805 
.830077 
.830349 
.830621 
.830893 

4.53 
4.53 
4.53 
4.53 

4  tq 

.170195 
.169923 
.169651 
.169379 
.169107 

57 
56 
55 
54 
53 

8 
9 
10 

.749056 
.749243 
.749429 

3.12 
3.10 
3.10 

.917891 
.917805 
.917719 

1.43 
1.43 
1.42 

.831165 
.831437 
.831709 

4.53 
4.53 
4.53 

.168835 
.168563 
.168291 

52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.749615 
.749801 
.749987 
.750172 
.750358 
.750543 
.750729 
.750914 
.751099 
.751284 

3.10 
3.10 
3.08 
3.10 
3.08 
3.10 
3.08 
3.08 
3.08 
3.08 

9.917634 
.917548 
.917462 
.917376 
.917290 
.917204 
.917118 
.917032 
.916946 
.916859 

1.43 
1.43 
1.43 
1.43 
1.43 
1.43 
1.43 
1.43 
1.45 
1.43 

9.831981 
.832253 
.832525 
.832796 
.833068 
.833339 
.833611 
.833882 
.834154 
.834425 

4.53 
4.53 
4.52 
4.53 
4.52 
4.53 
4.52 
4.53 
4.52 
4.52 

10.168019 
.167747 
.167475 
.167204 
.166932 
.166661 
.166389 
.166118 
.165846 
.165575 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.751469 
.751654 
.751839 
.752023 
.752208 
.752392 
.752576 
.752760 
.752944 
.753128 

3.08 
3.08 
3.07 
3.08 
3.07 
3.07 
3.07 
3.07 
3.07 
3.07 

9.916773 
.916687 
.916600 
.916514 
.916427 
.916341 
.916254 
.91616.7 
.916081 
.915994 

1.43 
1.45 
1.43 
1.45 
1.43 
.45 
.45 
.43 
.45 
.45 

9.834696 
.834967 
.835238 
.835509 
.835780 
.836051 
.836322 
.836593 
.836864 
.837134 

4.52 
4.52 
4.52 
4.52 
4.52 
4.52 
4.52 
4.52 
4.50 
4.52 

10.165304 
.165033 
.164762 
.164491 
.164220 
.163949 
.163678 
.163407 
.163136 
.162866 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.753312 
.753495 
.753679 
.753862 
.754046 
.754229 
.754412 
.754595 
.754778 
.754960 

3.05 
3.07 
3.07 
3.07 
3.05 
3.05 
3.05 
3.05 
3.03 
3.05 

9.915907 
.915820 
.915733 
.915646 
.915559 
.915472 
.915385 
.915297 
.915210 
.915123 

.45 
.45 
.45 
.45 
.45 
.45 
.47 
.45 
.45 
.47 

9.837405 
.837675 
.837946 
.838216 

.838487 
.838757 
.839027 
.839297 
.839568 
.839838 

4.50 
4.52 
4.50 
4.52 
4.50 
4.50 
4.50 
4.52 
4.50 
4.50 

10.162595 
.162325 
.162054 
.161784 
.161513 
.161243 
.160973 
.160703 
.160432 
.160162 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 

9.755143 
.755326 
.755508 
.755690 
.755872 
.756054 
.756236 
.756418 

3.05 
3.03 
3.03 
3.03 
3.03 
3.03 
3.03 

9.915035 
.914948 
.914860 
.914773 
.914685 
.914598 
.914510 
.914422 

.45 
.47 
.45 

.47 
.45 

.47 
.47 

9.840108 
.840378 
.840648 
.U0917 
.841187 
.841457 
.841727 
.841996 

4.50 
4.50 
4.48 
4.50 
4.50 
4.50 
4.48 

10.159892 
.159622 
.159352 
.159083 
.158813 
.158543 
.158273 
.158004 

19 
18 
17 
16 
15 
14 
13 
12 

49 
50 

.756600 
.756782 

3.03 
3.02 

.914334 
.914246 

.47 
.47 

.842266 
.842535 

4.48 
4.50 

.157734 
.157465 

11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.756963 
.757144 
.757326 
.757507 
.757688 
.757869 
.758050 
.758230 
.758411 
9.758591 

3.02 
303 
3.02 
3.02 
3.02 
3.02 
3.00 
3.02 
3.00 

9.914158 
.914070 
.913982 
.913894 
.913806 
.913718 
.913630 
913541 
.913453 
9.913365 

.47 
.47 
.47 
.47 
.47 
.47 
.48 
.47 
1.47 

9.842805 
.843074 
.843343 
.843612 
.843882 
.844151 
.844420 
.844689 
.844958 
9.845227 

4.48 
4.48 
4.48 
4.50 
4.48 
4.48 
4.48 
4.48 
4.48 

10.157195 
156926 
.156657 
.156388 
.156118 
.155849 
.155580 
.155311 
.155042 
10.154773 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  I'. 

Sine. 

D.  r. 

Cctang. 

D.I". 

Tang. 

' 

124* 


COSINES,   TANGENTS,  AND   COTANGENTS. 


l 

' 

Sine. 

D.  1*. 

Cosine. 

D.  1'. 

Tang. 

D.  r. 

»  Cotang. 

' 

0 

1 

2 
3 

9.758591 
.758772 
.758952 
.759132 

3.02 
3.00 
3.00 

9.913365 
.913276 
.913187 
.913099 

1.48 
1.48 
1.47 

1  4ft 

9.845227 
.845496 
.845764 
.846033 

4.48 
4.47 

4.48 

44ft 

10.154773 
.154504 
.154236 
.153967 

60 
59 

58 
57 

4 
5 

.759312 
.759492 

3.'00 

.913010 
.912922 

1  ,4O 

1.47 

14ft 

.846302 
.846570 

.48 

4.47 

.153698 
.153430 

56 
55 

6 

7 

.759672 
.759852 

3^00 

2QO 

.912833 
.912744 

.48 

1.48 

14ft 

.846839 
.847108 

4.48 
4.48 

.153161 
.152892 

54 
53 

8 
9 

.760031 
.760211 

.y8 
3.00 

2  QO 

.912655 
.912566 

.48 

1.48 

1  4ft 

.847376 
.847644 

4.47 
4.47 

4AQ. 

.152624 
.152356 

52 
51 

10 

.760390 

.y8 
2.98 

.912477 

1.48 

1.48 

.847913 

.48 

4.47 

.152087 

50 

11 

9.760569 

2QQ 

9.912388 

1  4ft 

9.848181 

4  47 

10.151819 

49 

12 
13 

.760748 
.760927 

.yo 

2.98 

.912299 
.912210 

1  .40 

1.48 

.848449 
.848717 

4^47 

.151551 
.151283 

48 
47 

14 

.761106 

2.  98 

2  no 

.912121 

.48 

.848986 

4.48 

.151014 

46 

15 

.761285 

.yo 

2  Oft 

.912031 

4ft 

.849254 

4.47 

447 

.150746 

45 

16 

.761464 

.  yo 

0  Q7 

.911942 

.40 
4Q 

.849522 

.4< 

.150478 

44 

17 
18 
19 
20 

.761642 
.761821 
.761999 
.762177 

<&,  J  1 

2.98 
2.97 
2.97 
2.98 

.911853 
.911763 
.911674 
.911584 

.48 

.50 
1.48 
1.50 
1.48 

.849790 
.850057 
.850325 
.850593 

4.47 
4.45 
4.47 
4.47 
4.47 

.150210 
.149943 
.149675 
.149407 

43 
42 

41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.762356 
.762534 
.762712 
.762889 
.763067 
.763245 
.763422 
.763600 
.763777 
.763954 

2.97 
2.97 
2.95 
2.97 
2.97 
2.95 
2.97 
2.95 
2.95 
2.95 

9.911495 
.911405 
.911315 
.911226 
.911136 
.911046 
.910956 
.910866 
.910776 
.910686 

1.50 
1.50 
1.48 
1.50 
1.50 
1.50 
1.50 
1.50 
1.50 
1.50 

9.850861 
.851129 
.851396 
.851664 
.851931 
.852199 
.852466 
.852733 
.853001 
.853268 

4.47 
4.45 
4.47 
4.43 
4.47 
4.45 
4.45 
4.47 
4.45 
4.45 

10.149139 
.148871 
.148604 
.148336 
.148069 
.147801 
.147534 
.147267 
.146999 
.146732 

39 
38 
37 
36 
35 
34 
35 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 

9.764131 
.764308 
.764485 
.764662 
.764838 
.765015 
.765191 

2.95 
2.95 
2.95 
2.93 
2.95 
2.93 

9.910596 
.910506 
.910415 
.910325 
.910235 
.910144 
.910054 

.50 
.52 
.50 
.50 
.52 
.50 

9.853535 
.853802 
.854069 
.854336 
.854603 
.854870 
.855137 

4.45 
4.45 
4.45 
4.45 
4.45 
4.45 

10.146465 
.146198 
.145931 
.145664 
.145397 
.145130 
.144863 

29 
28 
27 
26 
25 
24 
23 

38 
39 

.765367 
.765544 

2^95 

0  Q9, 

.909963 
.909873 

!so 

.855404 
.855671 

4.45 
4.45 

.144596 
.144329 

22 
21 

40 

.765720 

/v.  IfO 

2.93 

.909782 

:  ,52 

.855938 

4  .45 
4.43 

.144062 

20 

41 
42 
43 

44 
45 
46 
47 

48 
49 

9.765896 
.766072 
.766247 
.766423 
.766598 
.766774 
.766949 
.767124 
.767300 

2.93 
2.92 
2.93 
2.92 
2.93 
2.92 
2.92 
2.93 

2  no 

9.909691 
.909601 
.909510 
.909419 
.909328 
.909237 
.909146 
.909055 
.908964 

1.50 
1.52 
1.52 
1.52 
1.52 
1.52 
1.52 
1.52 
Ifn 

9.856204 
.856471 
.856737 
.857004 
.857270 
.857537 
.857803 
.858069 
.858336 

4.45 
4.43 
4.45 
4.43 
4.45 
4.43 
4.43 
4.45 

4  An 

10.143796 
.143529 
.143263 
.142996 
.142730 
.142463 
.142197 
.141931 
.141664 

19 
18 
17 
16 
15 
14 
13 
12 
11 

50 

.767475 

.  Vifl 

2.90 

.908873 

.060 

1.53 

.858602 

.4o 

4.43 

.141398 

10 

51 

9.767649 

0  Q0 

9.908781 

9.858868 

10.141132 

9 

52 
53 
54 
55 
56 
57 
58 
59 
60 

.767824 
.767999 
.768173 
.768348 
.768522 
.768697 
.768871 
.769045 
9.769219 

2.  '92 
2.90 
2.92 
2.90 
2.92 
2.90 
2.90 
2.90 

.908690 
.908599 
.908507 
.908416 
.908324 
.908233 
.908141 
.908049 
9.907958 

l!52 
1.53 
1.52 
1.53 
1.52 
1.53 
1.53 
1.52 

.859134 
.859400 
.859666 
.859932 
.860198 
.860464 
.860730 
.860995 
9.8612C1 

4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.42 
4.43 

.140866 
.140600 
.140334 
.140068 
.139802 
.139536 
.139270 
.139005 
10.138739 

8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  1". 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

' 

125' 


223 


54° 


TABLE   XII.      LOGARITHMIC    SIKES, 


' 

Sine. 

D.  1". 

Cosine. 

D.  1'. 

Tang. 

D.I". 

Cotang. 

> 

0 

1 

2 
3 

4 
5 

9.769219 
.769393 
.769566 
.769740 
.769913 
.770087 

2.90 

2.88 
2.90 
2.88 
2.90 

2OQ 

9.907958 
.907866 
.907774 
.907082 
.907590 
.907498 

1.53 
1.53 
1.53 
1.53 
1.53 

Ifrq 

9.861261 
.861527 
.861792 
.862058 
.862323 
.862589 

4.43 

4.42 
4.43 
4.42 
4.43 

10.138739 
.138473 
.138308 
.137942 
.137677 
.137411 

60 
59 
58 
57 
56 
55 

6 

7 
8 

.770260 
.770433 
.770606 

.OO 

2.88 
2.88 

.907406 
.907314 
.907222 

.OO 

1.53 
1.53 

.862854 
.863119 
.863385 

4.42 
4.42 
4.43 

.137146 
.136881 
.136615 

54 

53 
52 

9 
10 

.770779 
.770952 

2.'88 
2.88 

.907129 
.907037 

1  .55 
1.53 
1.53 

.863650 
.863915 

4.42 
4.42 
4.42 

.136350 
.136085 

51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.771125 
.771298 
.V71470 
.771643 
.771815 
.771987 
.772159 
.772331 
.772503 
.772675 

2.88 
2.87 
2.88 
2.87 
2.87 
2.87 
2.87 
2.87 
2.87 

9.906945 
.906852 
.906760 
.906667 
.906575 
.906482 
.906389 
.906296 
.906204 
.906111 

1.55 
1.53 
1.55 
1.53 
1.55 
1.55 
1.55 
1.53 
1.55 

9.864180 
.864445 
.864710 
.864975 
.865240 
.865505 
.865770 
.866035 
.866300 
.866564 

4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.40 

10.135820 
.135555 
.135290 
.135025 
.134760 
.134495 
.134230 
.133965 
.133700 
.133436 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

2.87 

1.55 

4.42 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.772847 
.773018 
.773190 
.773361 
.773533 
.773704 
.773875 
.774046 
.774217 
.774388 

2.85 
2.87 
2.85 
2.87 
2.85 
2.85 
2.85 
2.85 
2.85 
2.83 

9.906018 
.905925 
.905832 
.905739 
.905645 
.905552 
.905459 
.905366 
.905272 
.905179 

1.55 
1.55 
1.55 
1.57 
1.55 
1.55 
1.55 
1.57 
1.55 
1.57 

9.866829 
.867094 
.867358 
.867623 
.867887 
.868152 
.868416 
.868680 
.868945 
.869209 

4.42 
4.40 
4.42 
4.40 
4.42 
4.40 
4.40 
4  42 
4.40 
4.40 

10.133171 
.132906 
.132642 
.132377 
.132113 
.131848 
.131584 
.131320 
.131055 
.130791 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 

9.774558 
.774729 

2.85 

9.905085 
.904992 

1.55 

9.869473 
.869737 

4.40 

10.130527 
.130263 

29 

28 

33 
34 
35 

.774899 
.775070 
.775240 

2.83 
2.85 
2.83 

.904898 
.904804 
.904711 

1.57 
1.57 

1'55 

.870001 
.870265 
.870529 

4.40 
4.40 
4.40 

.129999 
.129735 
.129471 

27 
26 
25 

36 
37 
38 
39 

.775410 
.775580 
.775750 
.775920 

2.83 
2.83 
2.83 
2.83 

.904617 
.904523 
.904429 
.904335 

1.57 
1.57 
1.57 

1.57 

.870793 
.871057 
.871321 
.871585 

4.40 
4.40 
4.40 
4.40 

.129207 
.128943 
.128679 
.128415 

24 
23 
22 
21 

40 

.776090 

2.83 

.904241 

1  .57 
1.57 

.871849 

4.40 
4.38 

.128151 

20 

41 

9.776259 

2QQ 

9.904147 

1  K7 

9.872112 

10.127888 

19 

42 

.776429 

.OO 

.904053 

1  .O< 
1K7 

.872376 

A  ~4(\ 

.127624 

18 

43 

.776598 

00 

.903959 

.O< 

.872640 

4.40 

.127360 

17 

44 
45 

46 
47 
48 

.776768 
.776937 
.777106 
.777275 
.777444 

2.83 

2.82 
2.82 
2.82 
2.82 

.903864 
.903770 
.903676 
.903581 
.903487 

1  .58 
1.57 
1.57 
1.58 
1.57 

.872903 
.873167 
.873430 
.87'3694 
.873957 

4.38 
4.40 
4.38 
4.40 
4.38 

.127097 
.126833 
.126570 
.126306 
.126043 

16 
15 
14 
13 
12 

49 

.777613 

2.82 

.903392 

1  .58 

.874220 

4.38 

.125780 

11 

50 

.777781 

2.80 

.903298 

!  .57 

.874484 

4.40 

.125516 

10 

2.82 

4.38 

51 

9.777950 

2QO 

9.903203 

to 

9.874747 

4OQ 

10.125253 

9 

52 
53 
54 
55 

.778119 

.778287 
.778455 
.778624 

.04 

2.80 
2.80 
2.82 

.903108 
.9030M 
.902919 
.902824 

.  .00 

.57 
.58 
1.58 

.875010 
.875273 
.875537 
.875800 

.OO 

4.38 
4.40 
4.38 

4QQ 

.124990 
.124727 
.124463 
.124200 

8 
7 
6 
5 

56 
57 

.778792 
.778960 

2.80 
2.80 

.902729 
.902634 

1.58 

Ito 

.876063 
.876326 

.OO 

4.38 

4QQ 

.123937 
.123674 

4 
3 

58 
59 

.779128 
.779295 

2.80 
2.78 

.902539 
.902444 

.00 
1.58 

.876589 
.876852 

.00 
4.38 

.123411 
.123148 

2 

1 

60 

9.779463 

2.80 

9.902349 

1.58 

9.877114 

4.37 

10.122886 

0 

' 

Cosine. 

D.  1'. 

Sine. 

D.  1'. 

Cotang. 

D.  1". 

Tang. 

/ 

126° 


COSINES,  TANGENTS,   AND    COTANGENTS. 


' 

Sine. 

D.I.. 

Cosine. 

D.r. 

Tang. 

D.r. 

Cotang. 

' 

0 

1 

2 
3 
4 
5 
6 
7 

9.779463 
.779631 
.779798 
.779966 
.780133 
.780300 
.780467 
.780634 

2.80 

2.78 
2.80 
2.78 
2.78 
2.78 
2.78 

9.902349 
.902253 
.902158 
.902063 
.901967 
.901872 
.901776 
.901681 

1.60 
1.58 
1.58 
1.60 
1.58 
1.60 
1.58 

Art 

9.877114 

.877377 
.877640 
.877903 
.878165 
.878428 
.878691 
.878953 

4.38 
4.38 
4.38 
4.37 
4.38 
4.38 

4.37 
400 

10.122886 
.122623 
.122360 
.122097 
.121835 
.121572 
.121309 
.121047 

60 
59 
58 
57 
56 
55 
54 
53 

8 
9 

.780801 
.780968 

2.78 
2.78 

.901585 
.901490 

.OU 

.58 

Art 

.879216 
.879478 

.00 

4.37 
400 

.120784 
.120522 

52 
51 

10 

.781134 

2.77 
2.78 

.901394 

.OU 

.60 

.879741 

.00 
4.37 

.120259 

50 

11 
12 

9.781301 
.781468 

2.78 

9.901298 
.901202 

.60 

•   Cf\ 

9.880003 
.880265 

4.37. 

10.119997 
.119735 

49 

48 

13 

.781634 

2.77 

.901106 

.  OU 
Art 

.880528 

407 

.119472 

47 

14 
15 
16 

.781800 
.781966 

.782132 

2.77 

2.77 
2.77 

277 

.901010 
.900914 
.900818 

.OU 

.60 
.60 

IArt 

.880790 
.881052 
.881314 

.01 

4.3? 
4.37 

A  °.ft 

.119210 
.118948 
.118686 

46 
45 
44 

17 
18 
19 
20 

.782298 
.782464 
.782630 
.782796 

.  If 

2.77 
2.77 
2.77 
2.75 

.900722 
.900626 
.900529 
.900433 

.DU 

1.60 
1.62 
1.60 
1.60 

.881577 
.881839 
.882101 
.882363 

4  .00 

4.37 
4.37 
4.37 
4.37 

.118423 
.118161 
.117899 
.117637 

43 
42 
41 
40 

21 
22 
23 
24 
25 
26 

9.782961 
.783127 
.783292 
.783458 
.783623 
.783788 

2.77 
2.75 
2.77 
2.75 
2.75 

9.900337 
.900240 
.900144 
.900047 
.899951 
.899854 

1.62 

1.60 
1.62 
1.60 
1.62 

9.882625 
.882887 
.883148 
.883410 
.883672 
.883934 

4.37 
4.35 
4.37 
4.37 
4.37 
407 

10.117375 
.117113 
.116852 
.116590 
.116328 
.116066 

39 
38 
37 
36 
35 
34 

27 
28 
29 
30 

.783953 
.784118 
.784282 
.784447 

2.75 
2.75 
2.73 
2.75 
2.75 

.899757 
.899660 
.899564 
.899467 

1.62 
1.62 
1.60 
1.62 
1.62 

.884196 
.884457 
.884719 
.884980 

.O( 

4.35 
4.37 
4.35 
4.37 

.115804 
.115543 
.115281 
.115020 

33 
32 
31 
30 

31 
32 
33 

9.784612 
.784776 
.784941 

2.73 
2.75 

9.899370 
.899273 
.899176 

1.62 
1.62 

9.685242 
.885504 
.885765 

4.37 
4.35 

4  OK 

10.114758 
.114496 
.114235 

29 

28 
27 

34 
35 
36 
37 

38 

.785105 
.785269 
.785433 
.785597 
.785761 

2.73 
2.73 
2.73 
2.73 
2.73 

.899078 
.898981 
.898884 
.898787 
.898689 

1.63 
1.62 
1.62 
1.62 
1.63 

.886026 
.886288 
.886549 
.886811 
.887072 

.OO 

4.37 
4.35 
4.37 
4.35 

.113974 
.113712 
.113451 
.113189 
.112928 

26 
25 
24 
23 
22 

39 
40 

.785925 

.786089 

2.73 
2.73 

.898592 
.898494 

1.62 
1.63 

.887333 
.887594 

4.35 
4.35 

.112667 
.112406 

21 
20 

2.72 

1.62 

4.35 

41 

9.786252 

270 

9.898397 

IRQ 

9.887855 

4  OK 

10.112145 

19 

42 
43 
44 
45 

46 
47 
48 
49 
50 

.786416 
.786579 
.786742 
.786906 
.787069 
.787232 
.787395 
.787557 
.787720 

.  to 
2.72 
2.72 
2.73 
2.72 
2.72 
2.72 
2.70 
2.72 
2.72 

.898299 
.898202 
.898104 
.898006 
.897908 
.897810 
.897712 
.897614 
.897516 

.Do 

1.62 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 

.888116 
.888378 
.888639 
.888900 
.889161 
.889421 
.889682 
.889943 
.890204 

.oO 

4.37 
4.35 
4.35 
4.35 
4.33 
4.35 
4.35 
4.35 
4.35 

.111884 
.111622 
.111361 
.111100 
.110839 
.110579 
.110318 
.110057 
.109796 

18 
17 
16 
15 
14 
13 
12  1 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.787883 
.788045 
.788208 
.788370 
.788532 
.788694 
.788856 
.789018 
.789180 
9.789342 

2.70 
2.72 
2.70 
2.70 
2.70 
2.70 
2.70 
2.70 
2.70 

9.897418 
.897320 
.897222 
.897123 
.897025 
.896926 
.896828 
.896729 
.896631 
9.896532 

1.63 
1.63 
1.65 
1.63 
1.65 
1.63 
1.65 
1.63 
1.65 

9.890465 
.890725 
.890986 
.891247 
.891507 
.891768 
.892028 
.892289 
.892349 
9.892810 

4.33 
4.35 
4.35 
4.33 
4.35 
4.33 
4.35 
4.33 
4.35 

10.109535 
.109275 
.109014 
.108753 
.108493 
.108232 
.107972 
.107711 
.107451 
10.107190 

j 

6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D  I'. 

Sine. 

D.  1'. 

Cotang. 

D.r. 

Tang. 

1 

127° 


TABLE   XII.      LOGARITHMIC!   SIHES, 


141* 


a 

Sine. 

D.I'. 

Cosine. 

D.  r. 

Tang. 

D.I'. 

Cotang. 

' 

0 
1 

9.789342 
.789504 

2  70 

2  Aft 

9.896532 
.896433 

1.65 

1  AS 

9.892810 
.893070 

4.33 

4  OK 

10.107190 
.106930 

60 

59 

2 

.789665 

.  Do 

2  70 

.896335 

i  .00 

.893331 

.OO 
4qq 

.106669 

58 

3 
4 
5 
6 

.789827 
.789988 
.790149 
.790310 

2^68  ! 
2.68  ! 
2.68 

2  Aft 

.896236 
.896137 
.896038 
.895939 

lies 

1.65 
65 

893591 
.893851 
.894111 
.894372 

.00 
4.33 
4.33 
4.35 

4qq 

.106409 
.106149 
.105889 
.105628 

57 
56 
55 
54 

7 

.790471 

.  Do 
n  co 

.895840 

p.K 

.894632 

.Oo 

.105368 

53 

8 

.790632 

rf.Do 
2  Aft 

.895741 

.DO 

.894892 

4qq 

.105108 

52 

9 

.790793 

.Do 
2  Aft 

.895641 

•   CK 

.895152 

.00 

4OQ 

.104848 

51 

10 

.790954 

.  Do 

2.68 

.895542 

.DO 

.65 

.895412 

.00 

4.33 

.104588 

50 

11 
12 
13 
14 
'15 
16 

9.791115 
.791275 
.791436 
.791596 
.791757 
.791917 

2.67 
2.68 
2.67 
2.68 
2.67 

n  prr 

9.895443 
.895343 
.895244 
.895145 
.895045 
.894945 

.67 
.65 
.65 
.67 
.67 

A^ 

9.895672 
.895932 
.896192 
.896452 
.896712 
,896971 

4.33 
4.33 
4.33 
4.33 
4.32 

4qq 

10.104328 
.104068 
.103808 
.103548 
.103288 
.103029 

49 
48 
47 
46 
45 
44 

17 

18 

.792077 
.792237 

A.  D< 

2.67 

.894846 
.894746 

.DO 

:  .67 

.897231 
.897491 

.OO 

4.33 

.102769 
.102509 

43 
42 

19 
20 

.792397 
.792557 

2.67 
2.67 
2.65 

.894646 
.894546 

.67 
.67 
.67 

.897751 
.898010 

4.33 
4.32 
4.33 

.102249 
.101990 

41 
40 

21 
22 

9.792716 

.792876 

2.67 

n  OK 

9.894446 
.894346 

.67 

67 

9.898270 
.898530 

4.33 

4qp 

10.101730 
.101470 

39 
38 

23 
24 

25 
26 
27 
28 
29 
30 

.793035 
.793195 
.793354 
.793514 
.793673 
.793832 
.793991 
.794150 

<i  .  DO 

2.67 
2.65 
2.67 
2.65 
2.65 
2.65 
2.65 
2.63 

.894246 
.894146 
.894046 
.893946 
.893846 
.893745 
.893645 
.893544 

< 

.67 
.67 
.67 
.67 
.68 
.67 
.68 
.67 

.898789 
.899049 
.899308 
.899568 
.899827 
.900087 
.900346 
.900605 

.018 

4.33 
4.32 
4.33 
4.32 
4.33 
4.32 
4.32 
4.32 

.101211 
.100951 
.100692 
.100432 
.100173 
.099913 
.099654 
.099395 

37 
36 
35 
34 
33 
32 
31 
30 

31 

9.794308 

9.893444 

Aft 

9.900864 

4qq 

10.099136 

29 

32 

.794467 

O  JtK 

.893343 

.Do 

6rf 

.901124 

.OO 

4qp 

.098876 

23 

33 
34 

.794626 
.794784 

2.  '63 

2AQ 

.893243 
.893142 

( 

.68 

Aft 

.901383 
.901642 

.O<i 

4.32 

.098617 
.098358 

27 

26 

35 
36 

.794942 
.795101 

.  Do 

2.65 

.893041 
.892940 

.  Do 

.18 

.901901 
.902160 

4^32 

.098099 
.097840 

25 
24 

37 
38 

.795259 
.795417 

2.63 
2.63 

.892839 
.892739 

:  .68 
.67 

.902420 
.902679 

4.33 
4  32 

.097580 
.097321 

23 
22 

39 
40 

.795575 
.795733 

2.63 
2.63 
2.63 

.892638 
.892536 

'.  .68 
.70 
.68 

.902938 
.903197 

4.32 
4.32 
4.32 

.097062 
.096803 

21 
20 

41 

9.795891 

2  CO 

9.892435 

Aft 

9.903456 

A  Oft 

10.096544 

19 

42 

.796049 

.Do 

.892334 

.  .DO 

.903714 

4.  OU 

4qO 

.096286 

18 

43 

44 
45 

.796206 
.796364 
.796521 

2^63 
2.62 

2  CO 

.892233 
.892132 
.892030 

^68 
.70 

Aft 

.903973 
.904232 
.904491 

.  >)& 

4.32 
4.32 

4qo 

.096027 
.095768 
.095509 

17 
16 
15 

46 
47 

.796679 
.796836 

.DO 

2.62 

.891929 
.891827 

.Do 

.70 

Aft 

.904750 
.905008 

.0^ 

4.30 

4qp 

.095250 
.094992 

14 
13 

48 
49 

.796993 
.797150 

2^62 

2  on 

.891726 
.891624 

.Do 
Aft 

.905267 
.905526 

.09 

4.32 

.094733 
.094474 

12 
11 

50 

.797307 

.uB 

2.62 

.891523 

.Do 

.70 

.905785 

4^30 

.094215 

10 

51 

9.797464 

9.891421 

9.906043 

10.093957 

9 

52 
53 
54 
55 
56 

.797621 
.797777 
.797934 
.798091 
.798247 

2^60 
2.62 
2.62 
2.60 

.891319 
.891217 
.891115 
.891013 
.890911 

'.70 
.70 
.70 

:  .70 

.906302 
.906560 
.906819 
.907077 
.907336 

4^30 
4.32 
4.30 
4.32 

.093698 
.093440 
.093181 
.092923 
.092664 

8 
7 
6 
5 
4 

57 
58 
59 
60 

.798403 
.798560 
.798716 
9.798872 

2.60 
2.62 
2.60 
2.60 

.890809 
.890707 
.890605 
9.890503 

1  .70 
1.70 
1.70 
1.70 

^07594 
.907853 
.908111 
9.9083G9 

4^32 
4.30 
4.30 

.092406 
.092147 
.091889 
10.091631 

3 
2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  r.  i 

Cotang. 

D.  1". 

Tang. 

' 

39° 


COSINES,   TANGENTS,    AND    COTANGENTS. 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

9.798872 

2m 

9.890503 

9.908369 

10.091631  60 

1 

2 

.799028 
.799184 

.OU 
2.60 

.890400 
.890298 

l!70 

1  79 

.908628 
.908886 

4.32 
4.30 

4OA 

.091372 
.091114 

59 

58 

3 

.799339 

2.58 

.890195 

1  .  16 
1  7rt 

.909144 

.oU 

.090856  57 

4 
5 

.799495 
.799651 

2^60 

.890093 
.889990 

1  .  1  U 

1.72 

.909402 
.909660 

4^30 

.090598 
.090340 

56 
55 

6 

7 

.799806 
.799962 

2*.  60 

.889888 
.889785 

1.70 

1.72 

.909918 
.910177 

4.30 
4.32 

.090082 
.089823 

54 
53 

8 
9 
10 

.800117 
.800272 
.800427 

2.58 
2.58 
2.58 
2.58 

.889682 
.889579 
.889477 

1.72 
1.72 
1.70 
1.72 

.910435 
.910693 
.910951 

4.30 
4.30 
4.30 
4.30 

.089565 
.089307 
.089049 

52 
51 
50 

11 

12 

9.800582 
.800737 

2.58 

9.889374 
.889271 

1.72 

179 

9.911209 
.911467 

4.30 

4OA 

10.088791 
.088533 

49 

48 

13 
14 
15 
16 
17 

.800892 
.801047 
.801201 
.801356 
.801511 

2  '.58 
2.57 
2.58 
2.58 

.889168 
.889064 
.888961 
.888858 
.888755 

.  i<6 

1.73 
1.72 
1.72 
1.72 

.911725 
.911982 
.912240 
.912498 
.912756 

.oU 

4.28 
4.30 
4.30 
4.30 

.088275 
.088018 
.087760 
.087502 
.087244 

47 
46 
45 
44 
43 

18 
19 

.801665 
.801819 

2.57 
2.57 

.888651 
.888548 

1.73 

1.72 

.913014 
.913271 

4.30 
4.28 

.086986 
.086729 

42 
41 

20 

.801973 

2.57 
2.58 

.888444 

l!72 

.913529 

4.30 
4.30 

.086471 

40 

21 
22 
23 
24 

9.802128 
.802282 
.802436 
.802589 

2.57 
2.57 

2.55 

2fff 

9.888341 

.888237 
.888134 
.888030 

1.73 

1.72 

l-l? 

9.913787 
.914044 
.914302 
.914560 

4.28 
4.30 
4.30 

10.086213 
.085956 
.085698 
.085440 

39 
38 
37 
36 

25 

.802743 

.o< 

9  f^7 

.887926 

1.73 
1  ^ 

.914817 

4.28 

4OA 

.085183 

35 

26 

.802897 

<e.DY 

.887822 

1  .  to 

Ir-o 

.915075 

.O(J  • 
49Q 

.084925 

34 

27 
28 
29 

.803050 
.803204 
.803357 

2  .  55 
2.57 
2.55 

.887718 
.887614 
.887510 

.  to 

1.73 
1.73 

.915332 
.915590 
.915847 

.40 

4.30 

4.28 

.084668 
.084410 
.084153 

33 
32 
31 

30 

.803511 

2  .  •  '7 
2.55 

.887406 

1  .  73 
1.73 

.916104 

4.28 
4.30 

.083896 

30 

31 

9.803664 

9.887302 

9.916362 

49Q 

10.083638 

29 

32 
33 
34 
35 

.803817 
.803970 
.804123 
.804276 

2"  55 
2.55 
2.55 

2KK 

.887198 
.887093 
.886989 
.886885 

l!75 
1.73 
1.73 

.916619 
.916877 
.917134 
.917391 

.40 

4.30 
4.28 

4.28 

.083381 
.083123 
.082866 
.082609 

28 
27 
26 
25 

36 
37 
38 
39 

.804428 
.804581 
.804734 
.804886 

.OO 

2.55 
2.55 
2.53 

.886780 
.886676 
.886571 
.886466 

1.75 
1.73 
1.75 
1.75 

.917648 
.917906 
.918163 
.918420 

4.28 
4.30 
4.28 
4.28 

.082352 
.082094 
.081837 
.081580 

24 
23 
22 
21 

40 

.805039 

2.55 
2.53 

.886362 

1.73 
1.75 

.918677 

4.28 
4.28 

.081323 

20 

41 
42 
43 
44 
45 
46 
47 

9.805191 
.805343 
.805495 
.805647 
.805799 
.805951 
.806103 

2.53 
2.53 
2.53 
2.53 
2.53 
2.53 

SCO 

9.886257 
.886152 
.886047 
.885942 
.885837 
.885732 
.885627 

1.75 
1.75 
1.75 
1.75 
1.75 
1.75 

9.918934 
.919191 
.919448 
.919705 
.919962 
.920219 
.920476 

4.28 
4.28 
4.28 
4.28 
4.28 
4.28 

4QQ 

10.081066 
.080809 
.080552 
.080295 
.080038 
.079781 
.079524 

19 
18 
17 
16 
15 
14 
13 

48 

.806254 

.  0*2 

2KO 

.885522 

1  .75 

.920733 

.40 

.  079267 

12 

49 
50 

.806406 
.806557 

.  OO 

2.52 
2.53 

.885416 
.885311 

1.77 
1.75 
1.77 

.920990 
.921247 

4.28 
4.28 
4.27 

.079010 
.078753 

11 

10 

51 
52 

9.806709 
.806860 

2.52 

9.885205 
.885100 

1.75 

9.921503 

.921760 

4.28 

10.078497 
.078240 

9 

8 

53 

.807011 

2  53 

.884994 

1  .77 

.922017 

4.28 

49ft 

.077983 

7 

54 
55 
56 

58 
59 

.807163 
.807314 
.807465 
.807615 
.807766 
.807917 

2!  52 
2.52 
2.50 

2.52 
2.52 

2tA 

.884889 
.884783 
.884677 
:  884572 
.884466 
.884360 

1  .  75 
1.77 
1.77 
1.75 
1.77 
1.77 

.922274 
.922530 
.922787 
.923044 
.923300 
.923557 

,4O 

4.27 
4.28 
4.28 
4.27 
4.28 

.077726 
.077470 
.077213 
.076956 
.076700 
.076443 

6 
5 
4 
3 
2 
1 

60 

9.808067 

.ou 

9.884254 

1.77 

9.923814 

4.28 

10.076186 

0 

'  I  Cosine. 

D.  1". 

Sine.   D.  1*. 

Cotang. 

D.  1'. 

Tang. 

' 

129° 


TABLE   XII.      LOGARITHMIC   SISTES, 


' 

Sine. 

D.I'. 

Cosine. 

D.  1% 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

2 
3 

4 
5 
6 

9.808067 
.808218 
.808368 
.808519 
.808669 
.808819 
.808969 

2.52 
2.50 
2.52 
2.50 
2.50 
2.50 

2KA 

9.884254 

.884148 
.884042 
.883936 
.883829 
.883723 
.883617 

1.77 
1.77 
1.77 
1.78 
1.77 
1.77 

17ft 

9.923814 
.924070 
.924327 
.924583 
.924840 
.925096 
.925352 

4.27 
4.28 
4.27 
4.28 
4.27 
4.27 

10.076186 
.075930 
.075673 
.075417 
.075160 
.074904 
.074648 

60 
59 
58 
57 
56 
55 
54 

7 
8 

.809119 
.809269 

.OU 

2.50 
2  50 

.883510 
.883404 

.  to 

1.77 

1  7ft 

.925609 
.925865 

4.28 
4.27 

49ft 

.074391 
.074135 

53 
52 

9 

.809419 

.883297 

1  .  to 

177 

.926122 

.xo 

.078878 

51 

10 

.809569 

2^48 

.883191 

.  «  1 

1.78 

.926378 

4.27 
4.27 

.073622 

50 

11 
12 

9.809718 
.809868 

2.50 

2  48 

9.883084 
.882977 

1.78 

9.926634 
.926890 

4.27 

10.073366 
.073110 

49 

48 

13 
14 
15 

.810017 
.810167 
.810316 

2!50 
2.48 

.882871 
.882764 
.882657 

1.77 

.78 
.78 

O'ft 

.927147 
.927403 
.927659 

4.28 
4.27 
4.27 

.072853 
.072597 
.072341 

47 
46 
45 

16 

17 
18 
19 
20 

.810465 
.810614 
.810763 
.810912 
.811061 

2.  48 
2.48 
2.48 
2.48 
2.48 

.882550 
.882443 
.882336 
.882229 
.882121 

.  io 
.78 
.78 
1.78 
1.80 
1.78 

.927915 
.928171 
.928427 
.928684 
.928940 

4.27 
4.27 
4.27 
4.28 
4.27 
4.27 

.072085 
.071829 
.071573 
.071316 
.071060 

44 
43 
42 
41 
40 

21 
22 

23 
24 
25 
26 

27 
28 

9.811210 
.811358 
.811507 
.811655 
.811804 
.811952 
.812100 
.812248 

2.47 

2.48 
2.47 
2.48 
2.47 
2.47 
2.47 

9.882014 
.881907 
.881799 
.881692 
.881584 
.881477 
.881369 
.881261 

1.78 
'  1.80 
1.78 
1.80 
1.78 
1.80 
1.80 

9.929196 
.929452 
.929708 
.929964 
.930220 
.930475 
.930731 
.930987 

4.27 
4.27 
4.27 
4.27 
4.25 
4.27 
4.27 

10.070804 
.070548 
.070292 
.070036 
.069780 
.069525 
.069269 
.069013 

39 
38 
37 
36 
35 
34 
33 
32 

29 
30 

.812396 
.812544 

2.47 
2.47 
2.47 

.881153 
.881046 

1.80 
1.78 
1.80 

.931243 
.931499 

4.27 
4.27 
4.27 

.068757 
.068501 

31 
30 

31 

9.812692 

9  /<7 

9.880938 

Ion 

9.931755 

4  OK 

10.068245 

29 

32 
33 
34 
35 

.812840 
.812988 
.813135 
.813283 

2!47 
2.45 

2.47 

.880830 
.880722 
.880613 
.880505 

.oU 

1.80 
1.82 
1.80 

.932010 
.932266 
.932522 
.932778 

>«D 

4.27 
4.27 
4.27 

.067990 
.067734 
.067478 
.067222 

28 
27 
26 
25 

36 
37 

.813430 
.813578 

2.45 

2.47 

2AK 

.880397 
.880289 

1.80 
1.80 

1  ft9 

.933033 
.933289 

4.25 
4.27 

497 

.066967 
.066711 

24 
23 

38 

.813725 

.13 

2AK 

.880180 

1  .  Q/v 
10(\ 

.933545 

.ift 

.066455 

22 

39 
40 

.813872 
.814019 

.40 

2.45 

2.45 

.880072 
.879963 

.oO 
1.82 
1.80 

.933800 
.934056 

4.27 
4.25 

.066200 
.065944 

21 
20 

41 
42 
43 

9.814166 
.814313 
.814460 

2.45 
2.45 

9.879855 
.879746 
.879637 

1.82 
1.82 

9.934311 
.934567 
.934822 

4.27 
4.25 

10.065689 
.065433 
.065178 

19 
18 
17 

44 
45 
46 

.314607 
.814753 
.814900 

2.45 
2.43 
2.45 

.879529 
.879420 
.879311 

1.80 
1.82 
1.82 

1ft9 

.935078 
.935333 
.935589 

4.27 
4.25 

4.27 

.064922 
.064667 
.064411 

16 
15 
14 

47 
48 
49 
50 

.815046 
.815193 
.815339 
.815485 

2.43 
2.45 
2.43 
2.43 
2.45 

.879202 
.879093 
.878984 
.878875 

.O/i 

1.82 
1.82 
1.82 
1.82 

.935844 
.936100 
.936355 
.936611 

4^27 
4.25 
4.27 
4.25 

.064156 
.063900 
.063645 
.063389 

13 
12 
11 
10 

51 
52 

9.815632 
.815778 

2.43 

9.878766 
.878656 

1.83 

189 

9.936866 
.937121 

4.25 

497 

10.063134 
.062879 

9 

8 

53 
54 

.815924 
.816069 

2i42 

.878547 
.878438 

.04 

1.82 

1QQ 

.937377 
.937632 

.fSl 

4.25 

4  OK 

.062623 
.062368 

7 
6 

55 
56 

.816215 
.816361 

2.43 
2.43 

.878328 
.878219 

.OO 

1.82 

1QQ 

.937887 
.938142 

.3K) 

4.25 

497 

.062113 

.061858 

5 
4 

57 

.816507 

2.43 

.878109 

.OO 

.938398 

.4t 

.061602 

3 

58 
59 

.816652 
.816798 

2.42 
2.43 

.877999 
.877890 

1.83 
1.82 

.938653 
.938908 

4.25 
4.25 

.061347 
.061092 

1 

60 

9.816943 

2.42 

9.877780 

1.83 

9.939163 

4.25 

10.060837 

0 

' 

Cosine 

D.  1". 

Sine. 

D.  1'. 

Cotang. 

D.  r. 

Tang. 

' 

41° 


COSINES,   TANGENTS,   AI^D    COTANGENTS. 


' 

Sine. 

D.  1". 

Cosine. 

D.  1*. 

Tang. 

D.  r. 

Cotang. 

' 

0 

9.816943 

9.877780 

00 

9.939163 

10.060837 

60 

1 

.817088 

5*22 

.877670 

.Oo 

DO 

.939418 

Ant: 

.060582 

59 

2 
3 

.8172&3 
.817379 

2.4# 
2.43 

.877560 
.877450 

.Oo 
.83 

.939673 
.939928 

4  .6& 
4.25 

.060327 
.060072 

58 
57 

4 
5 

.817524 
.817668 

2.42 
2.40 

.877340 
.877230 

!83 

.940183 
.940439 

4  '.27 

.059817 
.059561 

56 
55 

6 

7 
8 

.817813 
.817958 
.818103 

2.42 
2.42 
2.42 

.877120 
.877010 
.876899 

.83 
.83 
.85 

DO 

.940694 
.940949 
.941204 

4.25 
4.25 
4.25 

4  OK 

.059306 
.059051 
.058796 

54 
53 

52 

9 

.818247 

o'^» 

.876789 

.  OO 

Q!r 

.  .941459 

.xo 

4OQ 

.058541 

51 

10 

.818392 

2.  $.6 
2.40 

.876678 

.OO 

.83 

.941713 

.6& 

4.25 

.058287 

50 

11 
13 

9.818536 

.818681 

2.42 

9.876568 
.876457 

.85 

DO 

9.941968 
.942223 

4.25 

10.058032 
.057777 

49 

48 

13 

.818825 

2.40 

.876347 

.  Oo 

.942478 

4.25 

.057522 

47 

14 
15 

.818939 
.819113 

2.40 
2.40 

.876236 
.876125 

.85 
.85 

.942733 
.942988 

4.25 
4^5 

.057267 
.057012 

46 
45 

16 

.819257 

2.40 

.876014 

.85 

oq 

.943243 

4.25 

.056757 

44 

17 
18 

.819401 
.819545 

2.40 

2.40 

.  87590  A 
.875793 

.Oo 

.85 

8K 

.943498 
.943752 

4.25 
4.23 

4  OK 

.056502 
.056248 

43 

42 

19 
20 

.819689 
.819832 

2.40 
2.38 
2.40 

.875682 
.875571 

D 

.85 
.87 

.944007 
.944262 

:.«D 

4.25 
4.25 

.055993 
.055738 

41 
40 

21 
22 
23 
24 
25 
26 

9.819976 
.820120 
.820263 
.820406 
.820550 
.820693 

2.40 
2.38 
2.38 
2.40 
2.38 

2OQ 

9.875459 
.875348 
.875237 
.875125 
.875014 
.874903 

.85 
.85 
1.85 

1.87 
1.85 

9.944517 
.944771 
.945026 
.945281 
.945535 
.945790 

4.23 
4.25 
4.25 
4.23 
4.25 

10.055483 
.055229 
.054974 
.054719 
.054465 
.054210 

39 
38 
37 
36 
35 
34 

27 

.820836 

.Oo 
9  Qft 

.874791 

1  PA 

.946045 

4oq 

.053955 

33 

28 
29 
30 

.820979 
.821122 
.821265 

/G.OO 

2.38 
2.38 
2.37 

.874680 
.874508 
.874456 

1  .OO 

1.87 
1.87 
1.87 

.946299 

.946554 
.946808 

.Xa 
4.25 
4.23 
4.25 

.053701 
.053446 
.053192 

32 
31 
30 

31 
32 
33 
34 

9.821407 
.821550 
.821693 

.821835 

2.38 
2.38 
2.37 

9.874344 

.874232 
.874121 
.874009 

1.87 
1.85 
1.87 

9.947063 
.947318 
.947572 
.947827 

4.25 
4.23 
4.25 

10.052937 
.052682 
.052428 
.052173 

29 
28 
27 
26 

35 

.821977 

2.37 

2  no 

.873898 

1.88 

.948081 

4.23 

4OQ 

.051919 

25 

36 
37 
38 
39 
40 

.822120 
.822202 
822404 
.822546 
.822688 

.OO 

2.37 
2.37 
2.37 
2.37 
2.37 

.873784 
.873672 
.873560 
.873448 
.873335 

1.87 
1.87 
1.87 
1.88 
1.87 

.948335 
.948590 
.948844 
.949099 
.949353 

.to 
4.25 
4.23 
4.25 
4.23 
4.25 

.051665 
.051410 
.051156 
.050901 
.050647 

24 
23 
22 
21 
20 

41 

9.822830 

9.873223 

CQ 

9.949608 

10.050392 

19 

42 
43 

.822972 
.823114 

2.37 
2.37 

2qe 

.873110 
.872998 

.00 
.87 

OQ 

.949862 
.950116 

4.23 
4.23 

.050138 
.049884 

18 
17 

44 

.823255 

.OO 

.872885 

.OO 

OQ 

.950371 

4oq 

.049629 

16 

45 

.823397 

207 

.8727?'2 

.  00 

QQ 

.950625 

.403 

400. 

.049375 

15 

46 

47 
48 
49 

.823539 

.823680 
.823821 
.823963 

.of 

2.35 
2.35 
2.37 

.872659 
.872547 
.872434 
.872321 

.OO 

.87 
.88 
1.88 

.950879 
.951133 
.951388 
.951642 

,<S9 

4.23 
4.25 
4.23 

4OQ 

.049121 

.048867 
.048612 
.048358 

14 
13 
12 
11 

50 

.824104 

2.35 

.872208 

1  .88 

.951896 

.in 

.048104 

10 

2.35 

1.88 

4.23 

51 
52 

9.824245 

.824386 

2.35 

2  OK 

9.872095 
.871981 

1.90 

1QQ 

9.952150 
.952405 

4.25 
400 

10.047850 
.047595 

9 

8 

53 
54 
55 

.824527 
.824668 
.824808 

.OO 

2.35 
2.33 

.871868 
.871755 
.871641 

.08 

1.88 
1.90 

.952659 
.952913 
.953167 

.60 

4.23 
4.23 

4OO 

.047341 
.047087 
.046833 

7 
6 
5 

56 
57 

.824949 
.825090 

2.35 
2.35 

200 

.871528 
.871414 

1  .88 
1.90 

.953421 
.953675 

.ISO 

4.23 
400. 

.046579 
.046325 

4 
3 

58 
59 
60 

.825230 
.825371 
9.825511 

.00 

2.35 
2.33 

.871301 
.871187 
9.871073 

1  .88 
1.90 
1.90 

.953929 
.954183 
9.954437 

.60 

4.23 
4.23 

.046071 
.045817 
10.045563 

2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  1'. 

Cctang. 

D.  1'. 

Tang. 

' 

181* 


2RQ 


TABLE   XII.      LOGARITHMIC   SINES,  137<l 


' 

Sine. 

D.r. 

Cosine. 

D.r. 

Tang. 

D.r. 

Cotang. 

i 

0 

9.825511 

2QQ 

9.871073 

1  ftQ 

9.954437 

10.045563 

60 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

.825651 
.825791 
.825931 
.826071 
.826211 
.826351 
.826491 
.826631 
.826770 
.826910 

.OO 

2.33 
2.33 
2.33 
2.33 
2.33 
2.33 
2.33 
2.32 
2.33 
2.32 

.870960 
.870846 
.870732 
.870618 
.870504 
.870390 
.870276 
.870161 
.870047 
.869933 

1  .00 

1.90 
1.90 
1.90 
1.90 
1.90 
1.90 
1.92 
1.90 
1.90 
1.92 

.954691 
.954946 
.955200 
.955454 
.955708 
.955961 
.956215 
.956469 
.956723 
.956977 

4  '.25 
4.23 
4.23 
4.23 
4.22 
4.23 
4.23 
4.23 
4.23 
4.23 

.045309 
.045054 
.044800 
.044546 
.044292 
.044039 
.043785 
.043531 
.043277 
.043023 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 

9.827049 

9.869818 

9.957231 

10.042769 

49 

12 
13 
14 
15 
16 
17 
18 
19 
20 

.827189 
.827328 
.827467 
.827606 
.827745 
.827884 
.828023 
.828162 
.828301 

2.o3 
2.32 
2.32 
2.32 
2.32 
2.32 
2.32 
2.32 
2.32 
2.30 

.869704 
.869589 
.869474 
.869360 
.869245 
.869130 
.869015 
.868900 
.868785 

1.90 
1.92 
1.92 
1.90 
1.92 
1.92 
1.92 
1.92 
1.92 
1.92 

.957485 
.957739 
.957993 
.958247 
.958500 
.958754 
.959008 
.959262 
.959516 

4.2o 
4.23 
4.23 
4.23 
4.22 
4.23 
4.23 
4.23 
4.23 
4.22 

.042515 
.042261 
.042007 
.041753 
.041500 
.041246 
.040992 
.040738 
.040484 

48 
47 
46 
45 
44 
43 
42 
41 
40 

21 

22 
23 
24 

25 
26 
27 

9.828439 
.828578 

.828716 
.828855 
.828993 
.829131 
.829269 

2.32 
2.30 
2.32 
2.30 
2.30 
2.30 

2OA 

9.868670  - 
.868555 
.868440 
.868324' 
.868209 
.868093 
.867978 

1.92 
1.92 
1.93 
1.92 
1.93 
1.92 

9.959769 
.960023 
.960277 
.960530 
.960784 
.961038 
.961292 

4.23 
4.23 
4.22 
4.23 
4.23 
4.23 
400 

10.040231 
.039977 
.039723 
.039470 
.039216 
.038962 
.038708 

39 
38 
37 
36 
35 
34 
33 

28 
29 

30 

.829407 
.829545 

.829683 

.O(} 

2.30 
2.30 
2.30 

.867862 
.867747 
.867631 

1^92 
1.93 
1.93 

.961545 
.961799 
.962052 

:.«« 

4.23 
4.22 
4.23 

.038455 
.038201 
.037948 

32 
31 
30 

31 
32 
33 
34 

D.  829821 
.829959 
.830097 
.830234 

2.30 
2.30 
2.28 

9.867515 
.867399 
.867283 
.867167 

1.93 
1.93 
1.93 

9.962306 
.962560 
.962813 
.963067 

4.23 
4.22 
4.23 

4->-> 

10.037694 
.037440 
.037187 
.036933 

29 
28 
27 
26 

35 
36 

37 

.830372 
.830509 
.830646 

2.30 
2.28 

2.28 

.867051 
.866935 
.866819 

1.93 
1.93 
1.93 

.963320 
.963574 
.963828 

.981 

4.23 
4.23 

.036680 
.036426 
.036172 

25 
24 
23 

38 
39 
40 

.830784 
.830921 
.831058 

2.30 

2.28 
2.28 
2.28 

.866703 
.866586 
.866470 

1.93 
1.95 
1.93 
1.95 

.964081 
.964335 
.964588 

4.22 
4.23 
4.22 
4.23 

.035919 
.035665 
.035412 

22 

21 
20 

41 
42 
43 
44 
45 
46 
47 
43 
4D 

id 

9.831195 
.831332 
.831469 
.831606 
.831742 
.831879 
.832015 
.832152 
.832288 
.832425 

2.28 
2.28 
2.28 
2.27 
2.28 
2.27 
2.28 
2.27 
2.28 
2.27 

9.866353 
.866237 
.866120 
.866004 
.865887 
.865770 
.865653 
.865536 
.865419 
.865302 

1.93 
1.95 
1.93 
1.95 
1.95 
1.95 
1.95 
1.95 
1.95 
1.95 

9.964842 
.965095 
.965349 
.965602 
.965855 
.966109 
.966362 
.966616 
.966869 
.967123 

4.22 
4.23 
4.22 
4.22 
4.23 
4.22 
4.23 
4.22 
4.23 
4.22 

10.035158 
.034905 
.034651 
.034398 
.034145 
.033891 
.033638 
.033384 
.033131 
.032877 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
62 
53 
54 
55 
56 
57 

9-832561 
.832697 
.832833 

^833105 
.833241 
.833377 

2.27 
2.27 

2.27 
2.27 
2.27 
2.27 

9.865185 
.865068 
.864950 

!  864716 
.864598 
.864481 

1.95 
1.97 
1.95 
.95 
.97 
.95 

9.967376 
.967629 
.967883 
.968136 
.968389 
.968643 
.968896 

4.22 
4.23 
4.22 
4.22 
4.23 
4.22 

10.032624 
.032371 
.032117 
.031864 
.031611 
.031357 
.031104 

9 

8 
7 
6 
5 
4 
3 

58 
59 
60 

.833512 
.833648 
9.833783 

2.25 
2.27 
2.25 

.864363 
.864245 
9.864127 

.97  i 
.97 
.97 

.969149 
.969403 
9.969656 

4.22 
4.23 

4.22 

.030851 
.030597 
10.030344 

2 
1 
0 

' 

Cosine. 

D.  1*. 

Sine. 

D.r. 

1  Cotang. 

D.r. 

Tang. 

' 

COSINES,  TANGENTS,  AND  COTAJNUuNTS. 


136° 


' 

Sine. 

D.I". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

9  .  833783 
.833919 

2.27 

29C 

9.864127 
.864010 

1.95 

1Q7 

9.969656 
.969909 

4.22 

499 

10.030344 
.030091 

60 
59 

2 
3 
4 

.834054 
.834189 
.834325 

.  ZO 

2.25 
2.27 

29K 

.  863892 
.863774 
.863656 

.  y  / 
1.97 
1.97 

1  Q7 

.970162 
.970416 
.970669 

.  ZZ 

4.23 
4.22 
4  22 

.029838 
.029584 
.029331 

58 
57 
56 

5 

6 

.834460 
.834595 

.  ZO 

2.25 
296 

.863538 
.863419 

i  .  y  / 
1.98 

1Q7 

.970922 
.971175 

4^22 

A  9Q 

.029078 
.028825 

55 
54 

7 
8 
9 

.834730 

.834865 
.834999 

.  zo 
2.25 
2.23 

29C 

.863301 
.863183 
.863064 

.  y  i 
1.97 
1.98 

1  07 

.971429 
.971682 
.971935 

%  .  ZO 

4.22 
4.22 
4  22 

.028571 
.028318 
.028065 

53 
52 
51 

10 

,835134 

.  ZO 

2.25 

.862946 

i  .  y  / 
1.98 

.972188 

4i22 

.027812 

50 

11 

9.835269 

2  23 

9.862827 

1  07 

9.972441 

4  23 

10.027559 

49 

12 

.835403 

29K 

.862709 

i  .  y  / 

1  QS 

.972695 

.027305 

48 

13 

.835538 

.  ZO 
290 

.862590 

i  .  ys 

.972948 

4  22 

.027052 

47 

14 

.835672 

.  Zo 
29C 

.862471 

1  "  Q7 

.973201 

.026799 

46 

15 

.835807 

.  ZO 
290 

.862353 

i  .y/ 

.973454 

499 

.026546 

45 

16 

.835941 

.  Zo 
290 

,862234 

1  *QQ 

.973707 

.  zz 
4  22 

.026293 

44 

17 
18 
19 

.836075 
.836209 
.836343 

.  Zo 

2.23 
2.23 
290 

.862115 
.861996 
.861877 

l!98 
1.98 

.973960 
.974213 
.974466 

4^22 
4.22 
4  23 

.026040 
.025787 
.025534 

43 

42 
41 

20 

.836477 

.  Zo 

2.23 

.861758 

2.'<X) 

.974720 

4^22 

.025280 

40 

21 

9.836611 

290 

9.861638 

9.974973 

499 

10.025027 

39 

22 

.836745 

.  zo 

299 

.861519 

1  .98 

.975226 

.  ZZ 
499 

.024774 

38 

23 
24 

.836878 
.837012 

.  ZZ 

2.23 

9  90 

.861400 
.861280 

1  .98 
2.00 

.975479 
.975732 

.  ZZ 

4.22 
4  22 

.024521 
.024268 

37 
36 

25 

26 

.837146 
.837279 

Z  .  ZO 

2.22 

299 

.861161 
.861041 

2.'  00 

.975985 
.976238 

4^22 

.024015 
.023762 

35 
34 

27 
28 
29 
30 

.837412 
.837546 
.837679 
.837812 

.  ZZ 

2.23 
2.22 

2.22 
2.22 

.860922 
.860802 
.860682 
.860562 

2^00 
2.00 
2.00 
2.00 

.976491 
.976744 
.976997 
.977250 

4!22 
4.22 
4.22 
4.22 

.023509 
.023256 
.023003 
.022750 

33 
32 
31 
30 

31 

9.837945 

290 

9.860442 

2(\(\ 

9.977503 

A  99 

10.022497 

29 

32 
33 
34 

.838078 
.838211 
.838344 

.  zz 
2.22 
2.22 
299 

.860322 
.860202 
.860082 

.  uu 
2.00 
2.00 
2nn 

.977756 
.978009 

.978262 

•*  .  zz 
4.22 
4.22 
4  22 

.022244 
.021991 
.021738 

28 
27 
26 

35 
36 

.838477 
.838610 

.  ZZ 

2.22 

.859962 
.859842 

.  uu 
2.00 

2ft9 

.978515 

.978768 

4^22 

499 

.021485 
.021232 

25 
24 

37 
38 
39 
40 

.838742 
.838875 
.839007 
.839140 

2!22 
2.20 
2.22 
2.20 

.859721 
.859601 
.859480 
.859360 

.  UZ 

2.00 
2.02 
2.00 
2.02 

.979021 
.979274 
.979527 
.979780 

.  ZZ 

4.22 
4.22 
4.22 

4.22 

.020979 
.020726 
.020473 
.020220 

23 

22 
21 
20 

41 

42 
43 
44 
45 

46 
47 
48 
49 

9.839272 
.839404 
.839536 
.839668 
.839800 
.839932 
.840064 
.840196 
.840328 

2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 

9.859239 
.859119 
.858998 
.858877 
.858756 
.858635 
.858514 
.858393 
.858272 

2.00 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 

9.980033 
.980286 
.980538 
.980791 
.981044 
.981297 
.981550 
.981803 
.982056 

4.22 
4.20 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

10.019967 
.019714 
.019462 
.019209 
.018956 
.018703 
.018450 
.018197 
.017944 

19 
18 
17 
16 
15 
14 
13 
12 
11 

50 

.840459 

2.  18 
2.20 

.858151 

2  .02 
2.03 

.982309 

4.22 
4.22 

.017691 

10 

51 
52 
53 
54 
55 
56 
57 
53 
59 
60 

9.840591 
.840722 
.840854 
.840985 
.841116 
.841247 
.841378 
.841509 
.841640 
9.841771 

2.18 
2.20 
2.18 
2.18 
2.18 
2.18 
2.18 
2.18 
2.18 

9.858029 
.  857908 
.857786 
.857665 
.857543 
.857422 
.857300 
.857178 
.857056 
9.856934 

2.02 
2.03 
2.02 
2.03 
2.02 
2.03 
2.03 
2.03 
2.03 

9.982562 
.982814 
.983067 
.983320 
.983573 
.983826 
.984079 
.984332 
.984584 
9.984837 

4.20 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.20 
4.22 

10.017438 
.017186 
.016933 
.016680 
.016427 
.016174 
.015921 
.015668 
.015416 
10.015163 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  I". 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

' 

133° 


241 


46° 


TABLE  XII.     LOGARITHMIC  SINES. 


135° 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  I". 

Cotang. 

t 

0 

1 

9.841771 
.841902 

2.18 

9  1R 

9.856934 
.856812 

2.03 

9  01 

9.984837 
.985090 

4.22 

499 

10.015163 
.014910 

60 
59 

2 
3 
4 

.842033 
.842163 
.842294 

z  .  lo 
2.17 
2.18 
2  17 

.856690 
.856568 
.856446 

Z  .  UO 

2.03 
2.03 

2AK 

.985343 
.985596 

.985848 

.  ZZ 

4.22 
4.20 

499 

.014657 
.014404 
.014152 

58 
57 
56 

5 

.842424 

2  18 

.856323 

.  UO 

2  03 

.986101 

.  ZZ 
499 

.013899 

55 

6 

.842555 

.856201 

2  AC 

.986354 

:  .  ZZ 

499 

.013646 

54 

7 
8 
9 

.842685 
.842815 
.842946 

2*17 
2.18 
2  17 

.856078 
.855956 
.855833 

.  UO 

2.03 
2.05 
2  03 

.986607 
.986860 
.987112 

.  ZZ 

4.22 
4.20 
4  22 

.013393 
.013140 

.012888 

53 
52 
51 

10 

.843076 

2.'  17 

.855711 

2.'05 

.987365 

4'22 

.012635 

50 

11 

9.843206 

217 

9.855588 

2  05 

9.987618 

4  22 

10.012382 

49 

12 

.843336 

.it 

21  7 

.855465 

.987871 

49O 

.012129 

48 

13 
14 

.843466 
.843595 

,il 

2.15 

217 

.855342 
.855219 

2.'05 
2  05 

.988123 
.988376 

.  ZU 

4.22 

4  99 

.011877 
.011624 

47 
46 

15 

.843725 

.  1  < 

2  17 

.855096 

2  05 

.988629 

t  .  ZZ 

4  22 

.011371 

45 

16 
17 

18 

.843855 
.843984 
.844114 

2*15 
2.17 
21  ^ 

.854973 
.854850 

.854727 

2*05 
2.05 

9  07 

.988882 
.989134 
.989387 

4^20 
4.22 

499 

.011118 
.010866 
.010613 

44 
43 

42 

19 

.844243 

.  10 

o  1  c 

.854603 

Z  .  Ui 

2  05 

.989640 

.  ZZ 
4  99 

.010360 

41 

20 

.844372 

Z  .  ID 

2.17 

.854480 

2.'  07 

989893 

1  .  ZZ 

4.20 

.010107 

40 

21 
22 

9.844502 
.844631 

2.15 

o  1  c 

9.854356 
.854233 

2.05 
2  07 

9.990145 
.990398 

4.22 

4  99 

10.009855 
.009602 

39 

38 

23 

.844760 

Z  .  IO 

OIK 

.854109 

2*05 

.990651 

t  .  ZZ 
490 

.009349 

37 

24 

.844889 

Z  .  IO 
21  K. 

.853986 

.990903 

.  ZU 
499 

.009097 

36 

25 

26 
27 

.845018 
.845127 
.845276 

.  ID 

2.15 
2.15 

21  t* 

.853862 
.853738 
.853614 

2i07 
2.07 

207 

.991156 
.991409 
.991662 

.  ZZ 

4.22 
4.22 

490 

.008844 
.008591 
.008338 

35 
34 
33 

28 

.845405 

.  IO 
21  3 

.853490 

.  Ui 

2  07 

.991914 

.  ZU 
4  99 

.008086 

32 

29 

.845533 

.  10 
o  i  c 

.853366 

2  07 

.992167 

1  .  ZZ 

4  22 

.  007833 

31 

30 

.845662 

Z  .,  IO 

2.13 

.853242 

2'07 

.992420 

4^20 

.007580 

30 

31 

9.845790 

0  1C 

9.853118 

9  07 

9.992672 

A  99 

10.007328 

29 

32 

.845919 

z  .  10 
2  13 

.852994 

z  .  u/ 
2  08 

.992925 

•*  .  ZZ 

4  22 

.007075 

28 

33 
34 
35 
36 

.846047 
.846175 
.846304 
.846432 

2  '.13 
2.15 
2.13 
21  ^ 

.852869 
.852745 
.852620 
.852496 

2  '.07 
2.08 
2.07 

.993178 
.993431 
.993683 
.993936 

4  '.22 
4.20 
4.22 
499 

.006822 
.006569 
.006317 
.006064 

27 
26 
25 
24 

37 
38 
39 

.846560 
.846688 
.846816 

.  10 

2.13 
2.13 
21  3 

.852371 
.852247 
.852122 

2!07 
2.08 

2  OR 

.994189 
.994441 
.994694 

.  zz 
4.20 

4.22 

4  99 

.005811 
.005559 
.005306 

23 
22 
21 

40 

.846944 

.  lo 
2.12 

.851997 

.  Uo 

2.08 

.994947 

t  .  ZZ 

4.20 

.005053 

20 

41 

9.847071 

21  ^ 

9.851872 

9  OR 

9.995199 

4  99 

10.004801 

19 

42 

.847199 

.  lo 
21  3 

.851747 

Z  .  UO 

9  OR 

.995452 

4  .  ZZ 

4  22 

.004548 

18 

43 
44 
45 

.847327 
.847454 

.847582 

.  10 

2.12 
2.13 

21  9 

.851622 
.851497 
.851372 

Z  .  Uo 

2.08 
2.08 

21  0 

.995705 
.995957 
.996210 

4!20 
4.22 

4  99 

.004295 
.004043 
.003790 

17 
16 
15 

46 

.847709 

.  1Z 

.851246 

.  1U 
2  OR 

.996463 

t  .  ZZ 
49O 

.003537 

14 

47 

.847836 

2  .  12 
21  <> 

.851121 

.Uo 
2  OR 

.996715 

.ZU 
499 

.003285 

13 

48 
49 
50 

.847964 
.848091 

.848218 

.  lo 
2.12 
2.12 
2.12 

.850996 
.850870 
.850745 

.  Uo 

2.10 
2.08 
2.10 

.996968 
.997221 
.997473 

.  ZZ 

4.22 
4.20 
4.22 

.003032 
.002779 
.002527 

12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 

9.848345 

.848472 
.848599 
.848726 
.848852 
.848979 
.849106 
.849232 
.849359 

2.12 
2.12 
2.12 
2.10 
2.12 
2.12 
2.10 
2.12 

9.850619 
.850493 
.850368 
.850242 
.850116 
.849990 
.849864 
.849738 
.849611 

2.10 
2.08 
2.10 
2.10 
2.10 
2.10 
2.10 
2.12 

9.997726 
.997979 
.998231 
.998484 
.998737 
.998989 
.999242 
.999495 
.999747 

4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 

499 

10.002274 
.002021 
.001769 
.001516 
.001263 
.001011 
.000758 
.000505 
.000253 

9 
8 
7 
6 
5 
4 
3 
2 
1 

60 

9.849485 

2  .  10 

9.849485 

2.  10 

10.000000 

.ZZ 

10.000000 

0 

' 

Cosine. 

D.  1". 

Sine. 

D.  I7. 

Cotang. 

D.  I'7. 

Tang. 

' 

134° 


242 


AZIMUTH   BY   ALTITUDE   OF   SUK. 


243 


ART.  42.    AZIMUTH  BY  ALTITUDE  OF  SUN. 

The  azimuth  of  a  given  line  may  be  determined  by  taking 
the  altitude  of  the  sun  with  an  engineer's  transit  having  a 
good  vertical  circle,  and  reading  the  horizontal  angle  between 
the  sun  and  the  line.  The  latitude  of  the  place  must  be 
known  and  a  nautical  almanac  must  be  at  hand  for  finding  the 
declination  of  the  sun  at  the  moment  of  observation. 

In  Fig  59  let  A  represent  the  center  of  the  celestial  sphere, 
Z  the  zenith,  P  the  pole,  N  the  north  point  of  the  horizon,  S 
the  position  of  the  sun  at  the  moment  of  observation.  Then, 
in  the  spherical  triangle  PZS,  the  angle  Z  is  the  azimuth  of 
the  sun,  and  this  is  the  same  as  the  horizontal  angle  NAC.  If 
AB  be  the  line  whose  azimuth  is  to  be  found,  NAB  is  its 
azimuth.  Now  if  the  horizontal  angle  BAG  be  measured,  and 
Z  be  computed,  the  azimuth  of  A  B  is  known. 

To  find  the  azimuth  of  the  sun  Z,  let  z  be  the  complement  of 
the  observed  altitude  CS,  corrected  for  refraction  and  parallax; 
let  <j>  be  the  latitude  of  the  place,  or  the  arc  NP;  let  d  be  the 
declination  of  the  sun,  or  the  arc  QS.  Then  in  the  spherical 
triangle  PZS  three  sides  are  known,  and  hence 


tan  JZ 


from  which  the  azimuth  Z  can  be  computed. 

In  the  figure  S  denotes  the  place  of  the  sun  in  the  summer 
half-year  when  £  is  positive, 
and  S'  its  place  in  the  winter 
half-year  when  d  is  negative. 
If  the  observation  be  made  in 
the  forenoon,  the  value  of  Z  is 
less  than  180  degrees;  if  it  be 
made  in  the  afternoon,  its  value 
is  greater  than  180  degrees. 

The  transit  having  been  put 
into  thorough  adjustment,  it  is 
set  up  at  A,  the  end  of  the  line 
AB,  whose  azimuth  is  to  be 
found.  The  vernier  of  the 
horizontal  limb  having  been  set  at  0°  00',  the  telescope  is 
pointed  at  B  and  the  alidade  undamped.  The  telescope  is 


FIG.  59. 


244 


AZIMUTH   BY   ALTITUDE   OF   SUK. 


then  pointed  upon  the  sun,  the  objective  and  eyepiece  being 
so  focused  that  the  shadow  of  the  cross-wires  and  the  image  of 
the  sun  may  be  plainly  seen  on  a  white  piece  of  paper  held 
behind  the  eyepiece.  The  cross-wires  should  be  made  tangent 
to  the  bright  circle  on  its  lower  and  right-hand  sides,  and  the 
horizontal  and  vertical  angles  be  read.  Next,  the  cross-wires 
should  be  made  tangent  on  the  upper  and  left-hand  sides  of 
the  bright  circle,  and  the  angles  be  read  again.  If  the  transit 
has  a  full  vertical  circle,  which  is  necessary  for  the  best  work, 
observations  should  be  taken  both  in  the  direct  and  reverse 
position  of  the  telescope. 

The  following  record  of  an  observation  will  illustrate  the 
method  of  making  the  measurements  and  obtaining  the  data 
for  computation.  The  declination  d  for  8:43  A.M.,  eastern 
standard  time,  of  the  day  of  observation,  is  here  taken  from  a 
nautical  almanac,  but  for  general  purposes  it  may  be  taken 


Time 
May  19, 
1897. 

Tel. 

Vertical 
Angle. 
CAS 

Horizontal 
Angle. 
BAG 

Data  and  Results. 

A.M. 

gh    4Qm 

42 

8     44 
46 

D 
R 

R 
D 

Wires  tang 
and  right 

43°  09'  00" 
43    35   30 

Wires  tang 
and  left 

44°  21'  00" 
44    48   00 

ent  to  lower 

sides. 

64°  48'  00" 
65    10   30 

ent  to  upper 
sides. 

64°  52'  30" 
65    15   00 

<£  =  40°  36'  27" 
5  at  7  A.M.  =  19°  53'  10" 
55 

5=19°  54'  05" 

Appar.  Alt.  =  43»  58'  22" 
Parallax  ...            +06 
Refraction..            —60 

Altitude  =  43°  57'  28" 
90    00   00 

2=46°  02'  32" 

Z=101°  45'  36" 
65    01    30 

Means  = 

43°  58'  22" 

65°  01'  30" 

NAB  =  36°  44'  06" 

from  the  solar  table  mentioned  on  page  126.  The  mean  ap- 
parent altitude  is  43°  58'  22",  and  this  being  corrected  for 
parallax  and  refraction,  the  zenith  distance  z  is  found.  By 
computation  from  the  formula,  the  mean  azimuth  of  the  sun  is 
101°  45'  36",  and  subtracting  from  this  the  mean  horizontal 
angle  BAC  the  final  azimuth  of  the  line  AB  is  36°  44'  06". 
The  uncertainty  of  an  azimuth  found  by  this  method  is  two 


MEAST   REFKACTIOK. 


<ar  tliree  minutes.  The  best  time  for  observation  is  when  the 
bearing  of  the  sun  is  nearly  east  or  nearly  west,  and  for  any 
precise  work  a  mean  result  should  be  determined  by  several 
morning  and  afternoon  observations. 

The  correction  for  parallax  of  the  sun  is  less  than  8". 6,  and 
is  always  added  to  the  apparent  altitude  ;  for  an  altitude  of  20° 
the  parallax  correction  is  8",  for  40°  it  is  7",  and  for  60°  it  is 
6".  In  precise  computations  the  value  of  the  parallax  cor* 
rection  may  be  found  by  multiplying  8". 6  by  the  cosine  of  the 
apparent  altitude  of  the  sun. 

The  correction  for  refraction  is  always  subtracted  from  the 
apparent  altitude,  and  its  value  is  to  be  taken  from  the  follow- 
ing table,  interpolating  when  necessary. 

TABLE  XIII.    MEAN  REFRACTIONS. 


a 

d 

. 

ej 

a 

0 

8 

.0 

0 

£  3 

0)  3 

§  3 

§  3 

33 

e§5 

o 

c 

eg 

i 

<8'S 

8 

££ 

8 

F 

Is 

£ 

2 

Ja 

§ 

0° 

i 

34'  54" 
24  25 

20° 
21 

2'  37" 
2  29 

40° 

41 

69" 
66 

60° 
61 

33" 
33 

2 

18  09 

22 

2  22 

42 

64 

62 

31 

3 

14  15 

23 

2  15 

43 

62 

63 

29 

4 

11  39 

24 

2  09 

44 

60 

64 

28 

5 

9  46 

25 

2  03 

45 

58 

65 

27 

6 

8  23 

26 

1  58 

46 

56 

66 

96 

7 

7  20 

27 

1  53 

47 

54 

67 

24 

3 

6  30 

28 

1  48 

48 

52 

68 

23 

9 

5  49 

29 

1  44 

49 

50 

69 

22 

10 

5  16 

30 

1  40 

50 

48 

70 

21 

11 

4  49 

31 

1  36 

51 

47 

72 

19 

12 

4  25 

32 

1  32 

52 

45 

74 

17 

13 

4  05 

33 

1  29 

53 

43 

76 

15 

14 

3  47 

34 

1  25 

54 

42 

78 

12 

15 

3  32 

35 

1  22 

55 

40 

80 

10 

16 

3  19 

36 

1  19 

56 

39 

82 

8 

17 

3  07 

37 

1  16 

57 

38 

84 

0 

18 

2  56 

38 

1  14 

58 

36 

86 

4 

19 

2  46 

39 

1  11 

59 

35 

88 

2 

20 

2  37 

40 

1  09 

60 

33 

90 

0 

346 


AREAS  AND   VOLUMES. 


AREAS  AND  VOLUMES. 


In  Fig.  60,  w  +  1  offsets, 
0i,  02,  .  ,  •  On+i,  distant  d 
apart,  are  measured  from  a 
line/*/  to  the  curved  boundary 
of  a  field  as  ab  .  .  .  bq.  Then 
the  area  of  abpqgf  is  given 
very  nearly  by  the  following 
formulas: 


FIG.  60. 


If  n  =  2, 
If  n  =  3, 
If  n  =  4, 
If  n  =  6, 


A  =  \d(0i  +  402  +  03)  (Simpson's  Rule). 

A  =  f<Z(Oi  +  302  +  303  +.04)         (Cotes'  Rule). 
^l  =  -/5^[7(0i  +  05)  +  32(02  +  04)  +  1203]. 
JL  =  -^[01+  03  +  05+07+5(02+  04+  06)  +  OJ 

(Weddles'  Rule). 
If  n  be  even, 

A  =  ld[0,  +  On+i  +4(02  +  .  .  .  +  0«)  +2(03  +.  .  .+  On-0], 
All  the  above  formulas  are  exact  if  the  curve  be  a  parabola  or 
a  straight  line. 

The  area  of  a  segment  of  a  circle  is,  very  nearly: 
A  = 


FIG.  61. 


-  (0.6  + 

This  formula  gives  areas  exact  to  five  places 
for  values  of  h  less  than  0.  Grand  a  maximum 
error,  when  h  =  r,  of  0.001  17r2.  For  more 
exact  results  when  h  =  0.6r,  0.7r,  0.8r,  0.9r, 
and  r,  use,  respectively,  0.6062,  0.  6076,  0.6089, 
0.6106,  and  0.6121  for  (0.6  +  O.OU/r)  in  the  formula. 

The  surface  of  a  segment  of  a  sphere  is  A  =  %irrh  (Fig.  61). 

Th«  volume  of  a  spherical  segment  is  (Fig.  61): 

V=   j7T^(3c2+7i2). 

If  a  solid  has  parallel  plane  ends  and  is  otherwise  bounded 
by  surfaces  that  can  be  generated  by  a  straight  line  always 
touching  the  peripheries  of  the  end  planes,  it  is  a  prismoid 
and  the  volume  is 


in  which  M  is  the  area  midway  between  the  end  areas  AI  and 
A2  and  I  is  the  distance  between  the  ends.  This  prismoidal 
formula  applies  also  to  spheres  and  ellipsoids.  It  is  widely 
used  for  the  computation  of  earthwork  volumes. 


I 

_LJ 

J U 


! 


1 

' 

I 

4- 

:  1 

—  [— 

:         - 

_L 

1  i 

±1 


rr 


H-H- 


4^+-i- 


4-H- 


1   i 


-H 


H-t- 


ttrr 


-M- 


1 

- 

. 

, 

1 

- 

-4-4 


E 


' 

. 

4 

•'. 

.. 

[ 

| 

| 

l 

[ 

s 
i 

4 

' 

I 

, 

•~- 

. 

o 

.' 

/' 

^ 

• 

- 

• 

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'  •                     ?  A  H.  Y 

SEP  4     1946 

'-y  1  9  2006 

VA  03088 


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